# Vector Algebra · Mathematics · JEE Main

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JEE Main 2024 (Online) 9th April Evening Shift
Between the following two statements: Statement I : Let $$\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$$ and $$\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$$. Then the ve...
JEE Main 2024 (Online) 9th April Evening Shift
Let $$\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers...
JEE Main 2024 (Online) 9th April Morning Shift
Let three vectors ,$$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overr... JEE Main 2024 (Online) 9th April Morning Shift Let$$\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}$$and$$\overrightarrow{O C}=3 \vec{b}$$, where$$O$$is the origin. If... JEE Main 2024 (Online) 8th April Evening Shift Let$$\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=11 \hat{i}-\hat{j}+\hat{k}$$and$$\overrightarrow{\mathrm{c}...
JEE Main 2024 (Online) 8th April Evening Shift
Let $$\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}+3 \hat{j}-5 \hat{k}$$ and $$\overrightarrow{\math... JEE Main 2024 (Online) 8th April Morning Shift The set of all$$\alpha$$, for which the vectors$$\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$$and$$\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}...
JEE Main 2024 (Online) 6th April Evening Shift
Let $$\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projectio...
JEE Main 2024 (Online) 6th April Evening Shift
Let $$\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$$. If $$\overrightarrow{\mathrm{c}}$$ ...
JEE Main 2024 (Online) 5th April Evening Shift
Let $$\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}$$ and $$\vec{c}$$ be three vectors such that $$(\vec{c}+\hat{i}) \tim... JEE Main 2024 (Online) 5th April Evening Shift Consider three vectors$$\vec{a}, \vec{b}, \vec{c}$$. Let$$|\vec{a}|=2,|\vec{b}|=3$$and$$\vec{a}=\vec{b} \times \vec{c}$$. If$$\alpha \in\left[0, ...
JEE Main 2024 (Online) 5th April Morning Shift
If $$\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)$$ and $$\mathrm{D}(-1,-4,-2)$$ are the vertices of a quadrilateral ABCD, then its are...
JEE Main 2024 (Online) 4th April Evening Shift
For $$\lambda>0$$, let $$\theta$$ be the angle between the vectors $$\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}$$ and $$\vec{b}=3 \hat{i}-\hat{j}+2 \ha... JEE Main 2024 (Online) 4th April Evening Shift Let$$\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$$and$$\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}$$. If$$\...
JEE Main 2024 (Online) 4th April Morning Shift
Let a unit vector which makes an angle of $$60^{\circ}$$ with $$2 \hat{i}+2 \hat{j}-\hat{k}$$ and an angle of $$45^{\circ}$$ with $$\hat{i}-\hat{k}$$ ...
JEE Main 2024 (Online) 1st February Morning Shift
Let $\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}$ and $\overrightarrow{\mathrm{... JEE Main 2024 (Online) 31st January Morning Shift Let $$\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$$ and $$\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$$ be three vectors. If a ve... JEE Main 2024 (Online) 31st January Morning Shift The distance of the point $$Q(0,2,-2)$$ form the line passing through the point $$P(5,-4, 3)$$ and perpendicular to the lines $$\vec{r}=(-3 \hat{i}+2 ... JEE Main 2024 (Online) 30th January Evening Shift Let$$\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$$. Let a vector$$\vec{b}$$be such that the angle between$$\vec{a}$...
JEE Main 2024 (Online) 30th January Evening Shift
Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}|=1$$ and $$|\vec{b} \times \vec{a}|=2$$. Then $$|(\vec{b} \times \vec{a})-\vec{b}|... JEE Main 2024 (Online) 30th January Morning Shift Let$$\overrightarrow{\mathrm{a}}=\mathrm{a}_1 \hat{i}+\mathrm{a}_2 \hat{j}+\mathrm{a}_3 \hat{k}$$and$$\overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat...
JEE Main 2024 (Online) 29th January Evening Shift
Let a unit vector $$\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}$$ make angles $$\frac{\pi}{2}, \frac{\pi}{3}$$ and $$\frac{2 \pi}{3}$$ with the vectors $$\f... JEE Main 2024 (Online) 29th January Evening Shift Let$$\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}$$, where O is the origin. If ... JEE Main 2024 (Online) 29th January Morning Shift Let$$\vec{a}, \vec{b}$$and$$\vec{c}$$be three non-zero vectors such that$$\vec{b}$$and$$\vec{c}$$are non-collinear. If$$\vec{a}+5 \vec{b}$$i... JEE Main 2024 (Online) 27th January Evening Shift Let the position vectors of the vertices$$\mathrm{A}, \mathrm{B}$$and$$\mathrm{C}$$of a triangle be$$2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{...
JEE Main 2024 (Online) 27th January Evening Shift
The position vectors of the vertices $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of a triangle are $$2 \hat{i}-3 \hat{j}+3 \hat{k}, 2 \hat{i}+2 \hat... JEE Main 2024 (Online) 27th January Morning Shift Let \overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k},  \overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k}). Let \overrightarrow{\mathrm{... JEE Main 2023 (Online) 15th April Morning Shift Let S be the set of all (\lambda, \mu) for which the vectors \lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k} and 3 \hat{i}-4 \ha... JEE Main 2023 (Online) 15th April Morning Shift Let \mathrm{ABCD} be a quadrilateral. If \mathrm{E} and \mathrm{F} are the mid points of the diagonals \mathrm{AC} and \mathrm{BD} respectiv... JEE Main 2023 (Online) 13th April Evening Shift Let$$|\vec{a}|=2,|\vec{b}|=3$$and the angle between the vectors$$\vec{a}$$and$$\vec{b}$$be$$\frac{\pi}{4}$$. Then$$|(\vec{a}+2 \vec{b}) \times...
JEE Main 2023 (Online) 13th April Evening Shift
Let for a triangle $$\mathrm{ABC}$$, $$\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}$$ $$\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta... JEE Main 2023 (Online) 13th April Morning Shift Let$$\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$$and$$\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$$. If a vector$$\vec{d}$... JEE Main 2023 (Online) 12th April Morning Shift Let $$a, b, c$$ be three distinct real numbers, none equal to one. If the vectors $$a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \h... JEE Main 2023 (Online) 12th April Morning Shift Let$$\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$$and$$\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$$. Let$$\vec{c}$$be a vector such ... JEE Main 2023 (Online) 11th April Evening Shift If four distinct points with position vectors$$\vec{a}, \vec{b}, \vec{c}$$and$$\vec{d}$$are coplanar, then$$[\vec{a} \,\,\vec{b} \,\,\vec{c}]$$i... JEE Main 2023 (Online) 11th April Morning Shift For any vector$$\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$$, with$$10\left|a_{i}\right| (A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\ri... JEE Main 2023 (Online) 11th April Morning Shift Let$$\vec{a}$$be a non-zero vector parallel to the line of intersection of the two planes described by$$\hat{i}+\hat{j}, \hat{i}+\hat{k}$$and$$\h... JEE Main 2023 (Online) 10th April Evening Shift Let $$\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$$ and $$\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{d}$$ be a vector which ... JEE Main 2023 (Online) 10th April Evening Shift If the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ are respectively the circumcenter and the orthocentre of a $$\triangle \mathrm{ABC}$$, then $$\overrig... JEE Main 2023 (Online) 10th April Morning Shift Let O be the origin and the position vector of the point P be$$ - \widehat i - 2\widehat j + 3\widehat k$$. If the position vectors of the points A, ... JEE Main 2023 (Online) 10th April Morning Shift An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If$$\overrightarrow {OP} = \overrightarrow u ,\overr... JEE Main 2023 (Online) 8th April Evening Shift Let the vectors $$\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}$$ and $$\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}$$ be copl... JEE Main 2023 (Online) 8th April Evening Shift The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)... JEE Main 2023 (Online) 8th April Morning Shift If the points with position vectors$$\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{... JEE Main 2023 (Online) 6th April Evening Shift Let the vectors $$\vec{a}, \vec{b}, \vec{c}$$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped... JEE Main 2023 (Online) 6th April Evening Shift The sum of all values of $$\alpha$$, for which the points whose position vectors are $$\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\al... JEE Main 2023 (Online) 6th April Morning Shift Let the position vectors of the points A, B, C and D be$$5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j... JEE Main 2023 (Online) 6th April Morning Shift Let $$\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$$ and $$\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$$. If $$\vec{d}$$ is a ... JEE Main 2023 (Online) 1st February Evening Shift Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is ... JEE Main 2023 (Online) 1st February Evening Shift Let $$\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$$ and $$\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$$ be three given vectors. If $$\overr... JEE Main 2023 (Online) 31st January Evening Shift Let \vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k} and \vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k} be three vectors. If \vec{r... JEE Main 2023 (Online) 31st January Morning Shift Let$$\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$$, and$$\vec{b}$$and$$\vec{c}$$be two nonzero vectors such that$$|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{... JEE Main 2023 (Online) 30th January Evening Shift Let$\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}$. If$((\vec{a}+\vec{b}) \times(\v...
JEE Main 2023 (Online) 30th January Evening Shift
Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec... JEE Main 2023 (Online) 30th January Morning Shift If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ are three non-zero vectors and $$\widehat n$$ is a unit vector perpendicular to $$\ove... JEE Main 2023 (Online) 30th January Morning Shift Let a unit vector$$\widehat{O P}$$make angles$$\alpha, \beta, \gamma$$with the positive directions of the co-ordinate axes$$\mathrm{OX}$$,$$\mat... JEE Main 2023 (Online) 29th January Evening Shift If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\wide... JEE Main 2023 (Online) 29th January Evening Shift Let$$\overrightarrow a = 4\widehat i + 3\widehat j$$and$$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If$$\overrightarrow c $$... JEE Main 2023 (Online) 29th January Morning Shift If the vectors$$\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$$,$$\overrightarrow b = - 2\widehat i + 4\widehat j - 2\wid... JEE Main 2023 (Online) 25th January Evening Shift Let $$\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1$$ and $$\overrightarrow a \times \ov... JEE Main 2023 (Online) 25th January Evening Shift If the four points, whose position vectors are$$3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \wideh... JEE Main 2023 (Online) 25th January Morning Shift The vector $$\overrightarrow a = - \widehat i + 2\widehat j + \widehat k$$ is rotated through a right angle, passing through the y-axis in its way a... JEE Main 2023 (Online) 25th January Morning Shift Let $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ be three non zero vectors such that $$\overrightarrow b$$ . $$\overrigh... JEE Main 2023 (Online) 24th January Evening Shift Let$$\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$$and$$\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$$. L... JEE Main 2023 (Online) 24th January Morning Shift Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that$${{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \ov... JEE Main 2023 (Online) 24th January Morning Shift Let $$\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\over... JEE Main 2022 (Online) 29th July Evening Shift Let$$\vec{a}, \vec{b}, \vec{c}$$be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magni... JEE Main 2022 (Online) 29th July Morning Shift Let$$\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}$$and$$\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}$$. Let$$\overrightarrow{\mathrm{c}}... JEE Main 2022 (Online) 29th July Morning Shift Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vect... JEE Main 2022 (Online) 28th July Evening Shift Let S be the set of all a $$\in R$$ for which the angle between the vectors $$\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$$ and $$\... JEE Main 2022 (Online) 28th July Morning Shift Let the vectors$$\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}$$and$$\vec{c}=t \hat{i}-t \hat{j}+\hat{... JEE Main 2022 (Online) 28th July Morning Shift Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}$$, the v... JEE Main 2022 (Online) 27th July Morning Shift Let $$\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$$ and $$\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b}=... JEE Main 2022 (Online) 27th July Morning Shift$$ \text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b... JEE Main 2022 (Online) 26th July Morning Shift Let $$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$$. If ... JEE Main 2022 (Online) 25th July Evening Shift Let $$\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$$ and $$\vec{a} \cdot \... JEE Main 2022 (Online) 25th July Morning Shift Let$$\mathrm{ABC}$$be a triangle such that$$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{... JEE Main 2022 (Online) 30th June Morning Shift Let a vector $$\overrightarrow c$$ be coplanar with the vectors $$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarro... JEE Main 2022 (Online) 29th June Evening Shift Let A, B, C be three points whose position vectors respectively are$$\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k\overrightarrow ... JEE Main 2022 (Online) 29th June Morning Shift Let $$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$, $$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$$ and$...
JEE Main 2022 (Online) 28th June Evening Shift
Let $$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$$...
JEE Main 2022 (Online) 28th June Evening Shift
Let $$\overrightarrow a$$ be a vector which is perpendicular to the vector $$3\widehat i + {1 \over 2}\widehat j + 2\widehat k$$. If $$\overrightarro... JEE Main 2022 (Online) 27th June Evening Shift Let$$\overrightarrow a $$and$$\overrightarrow b $$be the vectors along the diagonals of a parallelogram having area$$2\sqrt 2 $$. Let the angle b... JEE Main 2022 (Online) 27th June Morning Shift Let$$\overrightarrow a = \widehat i + \widehat j - \widehat k$$and$$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the numbe... JEE Main 2022 (Online) 26th June Morning Shift If$$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$and$$\overrightarrow c \,.\,\overrightarrow a =...
JEE Main 2022 (Online) 25th June Morning Shift
Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$ $${a_i} > 0$$, $$i = 1,2,3$$ be a vector which makes equal angles wit...
JEE Main 2022 (Online) 24th June Evening Shift
Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$$. If $$\theta... JEE Main 2022 (Online) 24th June Morning Shift Let$$\widehat a$$,$$\widehat b$$be unit vectors. If$$\overrightarrow c $$be a vector such that the angle between$$\widehat a$$and$$\overrighta...
JEE Main 2021 (Online) 31st August Evening Shift
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ three vectors mutually perpendicular to each other and have same magnitude. If a vect...
JEE Main 2021 (Online) 31st August Morning Shift
Let $$\overrightarrow a$$ and $$\overrightarrow b$$ be two vectors such that $$\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {... JEE Main 2021 (Online) 26th August Evening Shift A hall has a square floor of dimension 10 m$$\times$$10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is ... JEE Main 2021 (Online) 26th August Morning Shift Let$$\overrightarrow a = \widehat i + \widehat j + \widehat k$$and$$\overrightarrow b = \widehat j - \widehat k$$. If$$\overrightarrow c $$is a... JEE Main 2021 (Online) 27th July Evening Shift Let$$\overrightarrow a $$,$$\overrightarrow b $$and$$\overrightarrow c $$be three vectors such that$$\overrightarrow a $$=$$\overrightarrow b ...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the ve...
JEE Main 2021 (Online) 25th July Evening Shift
Let a, b and c be distinct positive numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k$$ and $$c\widehat i + c\wi... JEE Main 2021 (Online) 25th July Evening Shift If$$\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$$and$$\left| {\overrightarrow a \times \overrightarrow b } \ri...
JEE Main 2021 (Online) 25th July Morning Shift
Let the vectors$$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k$$ and $$(2 + b)\wid... JEE Main 2021 (Online) 22th July Evening Shift Let a vector$${\overrightarrow a }$$be coplanar with vectors$$\overrightarrow b = 2\widehat i + \widehat j + \widehat k$$and$$\overrightarrow c ...
JEE Main 2021 (Online) 22th July Evening Shift
Let three vectors $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ be such that $$\overrightarrow a \times \overrightarrow b... JEE Main 2021 (Online) 20th July Evening Shift In a triangle ABC, if$$\left| {\overrightarrow {BC} } \right| = 3$$,$$\left| {\overrightarrow {CA} } \right| = 5$$and$$\left| {\overrightarrow {BA...
JEE Main 2021 (Online) 20th July Morning Shift
Let $$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$. If $$\overrightarrow c$$ is...
JEE Main 2021 (Online) 18th March Evening Shift
Let $$\overrightarrow a$$ and $$\overrightarrow b$$ be two non-zero vectors perpendicular to each other and $$|\overrightarrow a | = |\overrightarro... JEE Main 2021 (Online) 18th March Evening Shift In a triangle ABC, if$$|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10$$, then the projection of the vector$$\o...
JEE Main 2021 (Online) 18th March Morning Shift
A vector $$\overrightarrow a$$ has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle...
JEE Main 2021 (Online) 17th March Evening Shift
Let O be the origin. Let $$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$$ and $$\overrightarrow {OQ} = - \widehat i + 2\widehat j ... JEE Main 2021 (Online) 17th March Morning Shift Let$$\overrightarrow a $$= 2$$\widehat i-$$3$$\widehat j$$+ 4$$\widehat k$$and$$\overrightarrow b $$= 7$$\widehat i$$+$$\widehat j-...
JEE Main 2021 (Online) 16th March Evening Shift
Let $$\overrightarrow a$$ = $$\widehat i$$ + 2$$\widehat j$$ $$-$$ 3$$\widehat k$$ and $$\overrightarrow b = 2\widehat i$$ $$-$$ 3$$\widehat j$$ + 5...
JEE Main 2021 (Online) 16th March Morning Shift
Let a vector $$\alpha \widehat i + \beta \widehat j$$ be obtained by rotating the vector $$\sqrt 3 \widehat i + \widehat j$$ by an angle 45$$^\circ$$ ...
JEE Main 2021 (Online) 26th February Evening Shift
If vectors $$\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$$ and $$\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat ... JEE Main 2021 (Online) 26th February Morning Shift If$$\overrightarrow a $$and$$\overrightarrow b $$are perpendicular, then$$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\ov...
JEE Main 2020 (Online) 5th September Morning Slot
If the volume of a parallelopiped, whose coterminus edges are given by the vectors $$\overrightarrow a = \widehat i + \widehat j + n\widehat k$$, $$\... JEE Main 2020 (Online) 4th September Morning Slot Let x0 be the point of Local maxima of$$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$, where$$\overright...
JEE Main 2020 (Online) 3rd September Evening Slot
Let a, b c $$\in$$ R be such that a2 + b2 + c2 = 1. If $$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\thet... JEE Main 2020 (Online) 3rd September Morning Slot The lines$$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$and$$\overrightarrow r = \...
JEE Main 2020 (Online) 8th January Evening Slot
Let $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ be two vectors. I...
JEE Main 2020 (Online) 8th January Morning Slot
Let the volume of a parallelopiped whose coterminous edges are given by $$\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$$, $$\over... JEE Main 2020 (Online) 7th January Evening Slot Let$$\overrightarrow a $$,$$\overrightarrow b $$and$$\overrightarrow c $$be three unit vectors such that$$\overrightarrow a + \vec b + \over...
JEE Main 2020 (Online) 7th January Morning Slot
A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vec...
JEE Main 2019 (Online) 12th April Evening Slot
Let $$\alpha$$ $$\in$$ R and the three vectors $$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\wide... JEE Main 2019 (Online) 12th April Morning Slot Let$$\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k$$and$$\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k$$be two vector... JEE Main 2019 (Online) 12th April Morning Slot If the volume of parallelopiped formed by the vectors$$\widehat i + \lambda \widehat j + \widehat k$$,$$\widehat j + \lambda \widehat k$$and$$\lam...
JEE Main 2019 (Online) 10th April Evening Slot
The distance of the point having position vector $$- \widehat i + 2\widehat j + 6\widehat k$$ from the straight line passing through the point (2, 3...
JEE Main 2019 (Online) 10th April Morning Slot
Let A (3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then cos ...
JEE Main 2019 (Online) 9th April Evening Slot
If a unit vector $$\overrightarrow a$$ makes angles $$\pi$$/3 with $$\widehat i$$ , $$\pi$$/ 4 with $$\widehat j$$ and $$\theta$$$$\in$$(0, $$\p... JEE Main 2019 (Online) 9th April Morning Slot Let$$\overrightarrow \alpha = 3\widehat i + \widehat j$$and$$\overrightarrow \beta = 2\widehat i - \widehat j + 3 \widehat k$$. If$$\overrigh...
JEE Main 2019 (Online) 8th April Evening Slot
Let $$\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge$$ and $$\mathop b\limits^ \to ... JEE Main 2019 (Online) 12th January Evening Slot Let$$\overrightarrow a $$,$$\overrightarrow b $$and$$\overrightarrow c $$be three unit vectors, out of which vectors$$\overrightarrow b $$and ... JEE Main 2019 (Online) 12th January Morning Slot The sum of the distinct real values of$$\mu $$, for which the vectors,$$\mu \widehat i + \widehat j + \widehat k,\widehat i + \mu \...
JEE Main 2019 (Online) 11th January Evening Slot
Let $$\sqrt 3 \widehat i + \widehat j,$$    $$\widehat i + \sqrt 3 \widehat j$$  and   $$\beta \widehat i + \l... JEE Main 2019 (Online) 11th January Morning Slot Let$$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,\overrightarrow b = \widehat i + \lambda \widehat j + 4\wideha...
JEE Main 2019 (Online) 10th January Evening Slot
If $$\overrightarrow \alpha$$ = $$\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b$$  and  $$\overrightarrow... JEE Main 2019 (Online) 10th January Morning Slot Let$$\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,\overrightarrow b = 4\widehat i + \left( {3 - {\la...
JEE Main 2019 (Online) 9th January Evening Slot
Let  $$\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$$   $$\overrightarrow b = {b_1}\widehat i + {b_2}\... JEE Main 2019 (Online) 9th January Morning Slot Let$$\overrightarrow a $$=$$\widehat i - \widehat j$$,$$\overrightarrow b $$=$$\widehat i + \widehat j + \widehat k$$and$$\overrightarrow c $$... JEE Main 2018 (Online) 16th April Morning Slot Let$$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow c = \widehat j - \widehat k$$and a vector$$\overrightarrow b $$be ... JEE Main 2018 (Offline) Let$$\overrightarrow u $$be a vector coplanar with the vectors$$\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k$$and$$\overrightarrow...
JEE Main 2018 (Online) 15th April Evening Slot
If the position vectors of the vertices A, B and C of a $$\Delta$$ ABC are respectively $$4\widehat i + 7\widehat j + 8\widehat k,$$   ...
JEE Main 2018 (Online) 15th April Morning Slot
If $$\overrightarrow a ,\,\,\overrightarrow b ,$$ and $$\overrightarrow C$$ are unit vectors such that $$\overrightarrow a + 2\overrightarrow b + ... JEE Main 2017 (Online) 9th April Morning Slot If the vector$$\overrightarrow b = 3\widehat j + 4\widehat k$$is written as the sum of a vector$$\overrightarrow {{b_1}} ,$$paralel to$$\overr...
JEE Main 2017 (Online) 8th April Morning Slot
The area (in sq. units) of the parallelogram whose diagonals are along the vectors $$8\widehat i - 6\widehat j$$ and $$3\widehat i + 4\widehat j - 12\... JEE Main 2017 (Offline) Let$$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$and$$\overrightarrow b = \widehat i + \widehat j$$. Let$$\overrightarrow c $$b... JEE Main 2016 (Online) 10th April Morning Slot Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are$$\overrightarrow a ,\overrightarrow b ,\overrightarrow...
JEE Main 2016 (Online) 9th April Morning Slot
In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3$$\widehat i$$ + $$\widehat j$$ $$-$$ $$\wide... JEE Main 2016 (Offline) Let$$\overrightarrow a ,\overrightarrow b $$and$$\overrightarrow c $$be three unit vectors such that$$\overrightarrow a \times \left( {\overrigh...
JEE Main 2015 (Offline)
Let $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ be three non-zero vectors such that no two of them are collinear and $$\left(... JEE Main 2014 (Offline) If$$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overri...
JEE Main 2013 (Offline)
If the vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k$$ and $$\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are the s...
AIEEE 2012
Let $$\overrightarrow a$$ and $$\overrightarrow b$$ be two unit vectors. If the vectors $$\,\overrightarrow c = \widehat a + 2\widehat b$$ and $$\o... AIEEE 2012 Let$$ABCD$$be a parallelogram such that$$\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p $$and$$\angle BAD$$... AIEEE 2011 The vectors$$\overrightarrow a $$and$$\overrightarrow b $$are not perpendicular and$$\overrightarrow c $$and$$\overrightarrow d $$are two vect... AIEEE 2011 If$$\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)$$and$$\overrightarrow b = {1 \over 7}\left( {2\widehat i...
AIEEE 2011
Let $$\overrightarrow a$$, $$\overrightarrow b$$, $$\overrightarrow c$$ be three non-zero vectors which are pairwise non-collinear. If $\overrighta... AIEEE 2010 Let $$\overrightarrow a = \widehat j - \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k.$$ Then the vector $$\overrightar... AIEEE 2010 If the vectors$$\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\... AIEEE 2009 If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w$$ are non-coplanar vectors and $$p,q$$ are real numbers, then the equality $$\left[ {3\o... AIEEE 2008 The vector$$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$$lies in the plane of the vectors$$\overrightarrow b = \wide... AIEEE 2008 The non-zero vectors are $${\overrightarrow a ,\overrightarrow b }$$ and $${\overrightarrow c }$$ are related by $${\overrightarrow a = 8\overrightar... AIEEE 2007 If$$\widehat u$$and$$\widehat v$$are unit vectors and$$\theta $$is the acute angle between them, then$$2\widehat u \times 3\widehat v$$is a un... AIEEE 2007 Let$$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k$$and$$\overrightarrow c ... AIEEE 2006 If $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \t... AIEEE 2006 The values of a, for which the points$$A, B, C$$with position vectors$$2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\wide... AIEEE 2005 If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then : AIEEE 2005 Let $$a, b$$ and $$c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k$$ and $$... AIEEE 2005 For any vector$${\overrightarrow a }$$, the value of$${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \tim... AIEEE 2005 If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ are non coplanar vectors and $$\lambda$$ is a real number then $$\left[ {\lambda \lef... AIEEE 2005 Let$$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1... AIEEE 2004 A particle acted on by constant forces $$4\widehat i + \widehat j - 3\widehat k$$ and $$3\widehat i + \widehat j - \widehat k$$ is displaced from the ... AIEEE 2004 Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w$$ be such that $$\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow ... AIEEE 2004 If$${\overrightarrow a ,\overrightarrow b ,\overrightarrow c }$$are non-coplanar vectors and$$\lambda $$is a real number, then the vectors$${\ove... AIEEE 2004 Let $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ be three non-zero vectors such that no two of these are collinear. If the vec... AIEEE 2004 Let $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ be non-zero vectors such that $$\left( {\overrightarrow a \times \overrighta... AIEEE 2003 If$$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$then... AIEEE 2003 Let$$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$and$$\overrightarrow w = \widehat i + 2\widehat ... AIEEE 2003 The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are th... AIEEE 2003 $$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c$$ are $$3$$ vectors, such that $$\overrightarrow a + \overrightarrow b + \overrighta... AIEEE 2003 A tetrahedron has vertices at$$O(0,0,0), A(1,2,1) B(2,1,3)$$and$$C(-1,1,2).$$Then the angle between the faces$$OAB$$and$$ABC$$will be : AIEEE 2003 If$$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr ... AIEEE 2003 Consider points $$A, B, C$$ and $$D$$ with position vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \wid... AIEEE 2003 If$$\overrightarrow u \,,\overrightarrow v $$and$$\overrightarrow w $$are three non-coplanar vectors, then$$\,\left( {\overrightarrow u + \overr... AIEEE 2002 If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrightarrow a }$$ and $${\overrigh... AIEEE 2002 If$$\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c $$are vectors such that$$\left[ {\overrightarrow a \,\overrightarrow b ... AIEEE 2002 If the vectors$\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$and$\overrightarrow{\mathbf{c}}$from the sides$B C, C A$and$A B$respec... AIEEE 2002 If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$$ thus what will be the v... AIEEE 2002 $$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vectors and $$\overrightarrow c$$ i... AIEEE 2002 If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \widehat j$$ are such that $$\ov... ## Numerical JEE Main 2024 (Online) 8th April Morning Shift Let$$\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$$and$$\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$$be three given ve... JEE Main 2024 (Online) 6th April Morning Shift Let$$\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$$. If$$x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$$for some$$x, y, z ... JEE Main 2024 (Online) 6th April Morning Shift Let $$\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$$ and a vector $$\vec{c}$$ be such that $$\vec{a} \times(\vec{b}+\v... JEE Main 2024 (Online) 5th April Morning Shift Let$$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$and$$\overrightarrow{\mathrm{c... JEE Main 2024 (Online) 4th April Morning Shift Let $$\mathrm{ABC}$$ be a triangle of area $$15 \sqrt{2}$$ and the vectors $$\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow... JEE Main 2024 (Online) 1st February Evening Shift Let \overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=-\hat{i}-8 \hat{j}+2 \hat{k} and \overrightarrow{\mathrm{c}}=4... JEE Main 2024 (Online) 31st January Evening Shift Let$$\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$$and$$\vec{c}$$be a vector such that$$(\vec{a}+\vec{b}) \times \vec... JEE Main 2024 (Online) 31st January Morning Shift Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=1,|\vec{b}|=4$$, and $$\vec{a} \cdot \vec{b}=2$$. If $$\vec{c}=(2 \vec{a} \times ... JEE Main 2024 (Online) 27th January Morning Shift The least positive integral value of \alpha, for which the angle between the vectors \alpha \hat{i}-2 \hat{j}+2 \hat{k} and \alpha \hat{i}+2 \alp... JEE Main 2023 (Online) 13th April Morning Shift Let$$\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$$and$$\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$$. If$$\vec{b}$$is a vector such that$$\vec{a}=\vec{b} \times... JEE Main 2023 (Online) 11th April Evening Shift Let $$\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$$ and $$\vec{b}=\hat{i}+\hat{j}-\hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c}=11, ... JEE Main 2023 (Online) 8th April Morning Shift Let$$\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$$and$$\vec{c}$$be vectors such that$$\vec{a} \times \vec... JEE Main 2023 (Online) 1st February Morning Shift Let $$\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{u}$$ be a vector such that $$|\vec{u}|=\alpha>... JEE Main 2023 (Online) 1st February Morning Shift$$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$$and$$D(4,5,0),|\lambda| \leq 5$$are the vertices of a quadrilateral$$A B C D$$. If its area is 18 square ... JEE Main 2023 (Online) 31st January Evening Shift Let \vec{a}, \vec{b}, \vec{c} be three vectors such that |\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2 and 2(\vec{a} \times \vec{b})=3(\vec{c} \tim... JEE Main 2023 (Online) 31st January Morning Shift Let$$\vec{a}$$and$$\vec{b}$$be two vectors such that$$|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$$and$$|\vec{a} \times \vec{b}|=\sqrt{48}$$. Then ... JEE Main 2023 (Online) 29th January Morning Shift Let$$\overrightarrow a $$,$$\overrightarrow b $$and$$\overrightarrow c $$be three non-zero non-coplanar vectors. Let the position vectors of four... JEE Main 2023 (Online) 24th January Evening Shift Let$$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrigh... JEE Main 2022 (Online) 29th July Evening Shift Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$$ and $$|\vec{a... JEE Main 2022 (Online) 27th July Evening Shift Let$$\overrightarrow a $$,$$\overrightarrow b $$,$$\overrightarrow c $$be three non-coplanar vectors such that$$\overrightarrow a \times$$... JEE Main 2022 (Online) 29th June Evening Shift Let$$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,$$\overrightarrow b = \widehat i + \widehat j + \widehat k$$&... JEE Main 2022 (Online) 28th June Morning Shift If$$\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$$,$$\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k$$and$$\overrightar... JEE Main 2022 (Online) 25th June Evening Shift Let $$\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$$, $$\lambda$$ $$\in$$ R. If $$\overrightarrow a$$ is a vector such that $$\o... JEE Main 2022 (Online) 25th June Morning Shift Let$$\theta$$be the angle between the vectors$$\overrightarrow a $$and$$\overrightarrow b $$, where$$|\overrightarrow a | = 4,|\overrightar... JEE Main 2021 (Online) 1st September Evening Shift Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector$...
JEE Main 2021 (Online) 27th August Morning Shift
Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$$ and $$... JEE Main 2021 (Online) 26th August Evening Shift If the projection of the vector$$\widehat i + 2\widehat j + \widehat k$$on the sum of the two vectors$$2\widehat i + 4\widehat j - 5\widehat k$$an... JEE Main 2021 (Online) 27th July Evening Shift Let$$\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$$,$$\overrightarrow b = 3\widehat i + \beta \widehat j - \a...
JEE Main 2021 (Online) 27th July Morning Shift
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b$$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vec...
JEE Main 2021 (Online) 25th July Evening Shift
If $$\left( {\overrightarrow a + 3\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 5\overrightarrow b } \right)$$ an...
JEE Main 2021 (Online) 25th July Morning Shift
Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors....
JEE Main 2021 (Online) 20th July Evening Shift
For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $${\overrightarrow v _1} = \sqr... JEE Main 2021 (Online) 20th July Morning Shift Let$$\overrightarrow a $$,$$\overrightarrow b $$,$$\overrightarrow c $$be three mutually perpendicular vectors of the same magnitude and equally i... JEE Main 2021 (Online) 20th July Morning Shift If the shortest distance between the lines$$\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\wideha...
JEE Main 2021 (Online) 17th March Evening Shift
Let $$\overrightarrow x$$ be a vector in the plane containing vectors $$\overrightarrow a = 2\widehat i - \widehat j + \widehat k$$ and $$\overright... JEE Main 2021 (Online) 17th March Morning Shift If$$\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$$,$$\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \wideh...
JEE Main 2021 (Online) 16th March Evening Shift
Let $$\overrightarrow c$$ be a vector perpendicular to the vectors, $$\overrightarrow a$$ = $$\widehat i$$ + $$\widehat j$$ $$-$$ $$\widehat k$$ and...
JEE Main 2021 (Online) 25th February Evening Shift
Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. I...
JEE Main 2021 (Online) 25th February Morning Shift
Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and $$\overrightarrow c = \wideh... JEE Main 2021 (Online) 24th February Morning Shift Let three vectors$$\overrightarrow a ,\overrightarrow b $$and$$\overrightarrow c $$be such that$$\overrightarrow c $$is coplanar with$$\ov...
JEE Main 2020 (Online) 6th September Evening Slot
If $$\overrightarrow x$$ and $$\overrightarrow y$$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \l... JEE Main 2020 (Online) 6th September Morning Slot If$$\overrightarrow a $$and$$\overrightarrow b $$are unit vectors, then the greatest value of$$\sqrt 3 \left| {\overrightarrow a + \overrightarr...
JEE Main 2020 (Online) 5th September Evening Slot
Let the vectors $$\overrightarrow a$$, $$\overrightarrow b$$, $$\overrightarrow c$$ be such that $$\left| {\overrightarrow a } \right| = 2$$, $$\l... JEE Main 2020 (Online) 4th September Evening Slot If$$\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$$, then the value of$${\left| {\widehat i \times \left( {\overrightarrow a \times ...
JEE Main 2020 (Online) 2nd September Evening Slot
Let the position vectors of points 'A' and 'B' be $$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + \widehat j + 3\widehat k$$, respectivel...
JEE Main 2020 (Online) 2nd September Morning Slot
Let $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ be three unit vectors such that $${\left| {\overrightarrow a - \overrig... JEE Main 2020 (Online) 9th January Evening Slot Let$$\overrightarrow a $$,$$\overrightarrow b $$and$$\overrightarrow c $$be three vectors such that$$\left| {\overrightarrow a } \right| = \sqrt...
JEE Main 2020 (Online) 9th January Morning Slot
If the vectors, $$\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k$$, \overrightarrow q = a\widehat i + \left( {a...
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