Vector Algebra · Mathematics · JEE Main

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...

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...

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...

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...

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}...

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...

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}...

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...

Let $$\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$$. If $$\overrightarrow{\mathrm{c}}$$ ...

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...

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, ...

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...

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...

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 $$\...

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 $$\hat{i}-\hat{k}$$ ...

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{...

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...

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 ...

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}$...

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}|...

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...

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...

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 ...

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...

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{...

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...

Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $ $\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\overrightarrow{\mathrm{...

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...

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...

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...

Let for a triangle $$\mathrm{ABC}$$, $$\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}$$ $$\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta...

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}$...

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...

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 ...

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...

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...

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...

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 ...

If the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ are respectively the circumcenter and the orthocentre of a $$\triangle \mathrm{ABC}$$, then $$\overrig...

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, ...

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...

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...

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)$...

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{...

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...

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...

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...

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 ...

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 ...

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...

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...

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{...

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...

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...

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are three non-zero vectors and $$\widehat n$$ is a unit vector perpendicular to $$\ove...

Let a unit vector $$\widehat{O P}$$ make angles $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes $$\mathrm{OX}$$, $$\mat...

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\wide...

Let $$\overrightarrow a = 4\widehat i + 3\widehat j$$ and $$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If $$\overrightarrow c $$...

If the vectors $$\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$$, $$\overrightarrow b = - 2\widehat i + 4\widehat j - 2\wid...

Let $$\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1$$ and $$\overrightarrow a \times \ov...

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...

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...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three non zero vectors such that $$\overrightarrow b $$ . $$\overrigh...

Let $$\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$$ and $$\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$$. L...

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...

Let $$\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\over...

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...

Let $$\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}$$. Let $$\overrightarrow{\mathrm{c}}...

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...

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 $$\...

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{...

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...

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}=...

$$ \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...

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 ...

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 \...

Let $$\mathrm{ABC}$$ be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{...

Let a vector $$\overrightarrow c $$ be coplanar with the vectors $$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarro...

Let A, B, C be three points whose position vectors respectively are $$\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k$$ $$\overrightarrow ...

Let $$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$, $$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$$ and $...

Let $$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$$...

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...

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be the vectors along the diagonals of a parallelogram having area $$2\sqrt 2 $$. Let the angle b...

Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the numbe...

If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $$\overrightarrow c \,.\,\overrightarrow a =...

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...

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...

Let $$\widehat a$$, $$\widehat b$$ be unit vectors. If $$\overrightarrow c $$ be a vector such that the angle between $$\widehat a$$ and $$\overrighta...

Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ three vectors mutually perpendicular to each other and have same magnitude. If a vect...

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two vectors such that $$\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {...

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 $...

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = \widehat j - \widehat k$$. If $$\overrightarrow c $$ is a...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\overrightarrow a $$ = $$\overrightarrow b ...

Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the ve...

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...

If $$\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$$ and $$\left| {\overrightarrow a \times \overrightarrow b } \ri...

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...

Let a vector $${\overrightarrow a }$$ be coplanar with vectors $$\overrightarrow b = 2\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c ...

Let three vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow a \times \overrightarrow b...

In a triangle ABC, if $$\left| {\overrightarrow {BC} } \right| = 3$$, $$\left| {\overrightarrow {CA} } \right| = 5$$ and $$\left| {\overrightarrow {BA...

Let $$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$. If $$\overrightarrow c $$ is...

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two non-zero vectors perpendicular to each other and $$|\overrightarrow a | = |\overrightarro...

In a triangle ABC, if $$|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10$$, then the projection of the vector $$\o...

A vector $$\overrightarrow a $$ has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle...

Let O be the origin. Let $$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$$ and $$\overrightarrow {OQ} = - \widehat i + 2\widehat j ...

Let $$\overrightarrow a $$ = 2$$\widehat i$$ $$-$$ 3$$\widehat j$$ + 4$$\widehat k$$ and $$\overrightarrow b $$ = 7$$\widehat i$$ + $$\widehat j$$ $$-...

Let $$\overrightarrow a $$ = $$\widehat i$$ + 2$$\widehat j$$ $$-$$ 3$$\widehat k$$ and $$\overrightarrow b = 2\widehat i$$ $$-$$ 3$$\widehat j$$ + 5...

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$$^\circ$$ ...

If vectors $$\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$$ and $$\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat ...

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are perpendicular, then $$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\ov...

If the volume of a parallelopiped, whose coterminus edges are given by the vectors $$\overrightarrow a = \widehat i + \widehat j + n\widehat k$$, $$\...

Let x0 be the point of Local maxima of $$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$, where $$\overright...

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...

The lines $$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$ and $$\overrightarrow r = \...

Let $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ be two vectors. I...

Let the volume of a parallelopiped whose coterminous edges are given by $$\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$$, $$\over...

Let $$\overrightarrow a $$ , $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $$\overrightarrow a + \vec b + \over...

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...

Let $$\alpha $$ $$ \in $$ R and the three vectors $$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\wide...

If the volume of parallelopiped formed by the vectors $$\widehat i + \lambda \widehat j + \widehat k$$, $$\widehat j + \lambda \widehat k$$ and $$\lam...

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...

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...

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 ...

If a unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$ , $$\pi $$/ 4 with $$\widehat j$$ and $$\theta $$$$ \in $$(0, $$\p...

Let $$\overrightarrow \alpha = 3\widehat i + \widehat j$$ and $$\overrightarrow \beta = 2\widehat i - \widehat j + 3 \widehat k$$ . If $$\overrigh...

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 ...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors, out of which vectors $$\overrightarrow b $$ and ...

The sum of the distinct real values of $$\mu $$, for which the vectors, $$\mu \widehat i + \widehat j + \widehat k,$$   $$\widehat i + \mu \...

Let $$\sqrt 3 \widehat i + \widehat j,$$    $$\widehat i + \sqrt 3 \widehat j$$  and   $$\beta \widehat i + \l...

Let  $$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,$$ $$\overrightarrow b = \widehat i + \lambda \widehat j + 4\wideha...

If $$\overrightarrow \alpha $$ = $$\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b $$  and  $$\overrightarrow...

Let $$\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,$$   $$\overrightarrow b = 4\widehat i + \left( {3 - {\la...

Let  $$\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$$   $$\overrightarrow b = {b_1}\widehat i + {b_2}\...

Let $$\overrightarrow a $$ = $$\widehat i - \widehat j$$, $$\overrightarrow b $$ = $$\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c $$...

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow c = \widehat j - \widehat k$$ and a vector $$\overrightarrow b $$ be ...

Let $$\overrightarrow u $$ be a vector coplanar with the vectors $$\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k$$ and $$\overrightarrow...

If the position vectors of the vertices A, B and C of a $$\Delta $$ ABC are respectively $$4\widehat i + 7\widehat j + 8\widehat k,$$   ...

If $$\overrightarrow a ,\,\,\overrightarrow b ,$$ and $$\overrightarrow C $$ are unit vectors such that $$\overrightarrow a + 2\overrightarrow b + ...

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...

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\...

Let $$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$. Let $$\overrightarrow c $$ b...

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...

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...

Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $$\overrightarrow a \times \left( {\overrigh...

Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of them are collinear and $$\left(...

If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overri...

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...

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two unit vectors. If the vectors $$\,\overrightarrow c = \widehat a + 2\widehat b$$ and $$\o...

Let $$ABCD$$ be a parallelogram such that $$\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p $$ and $$\angle BAD$$...

The vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ are not perpendicular and $$\overrightarrow c $$ and $$\overrightarrow d $$ are two vect...

If $$\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)$$ and $$\overrightarrow b = {1 \over 7}\left( {2\widehat i...

Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-zero vectors which are pairwise non-collinear. If $\overrighta...

Let $$\overrightarrow a = \widehat j - \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k.$$ Then the vector $$\overrightar...

If the vectors $$\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\...

If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ are non-coplanar vectors and $$p,q$$ are real numbers, then the equality $$\left[ {3\o...

The vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$$ lies in the plane of the vectors $$\overrightarrow b = \wide...

The non-zero vectors are $${\overrightarrow a ,\overrightarrow b }$$ and $${\overrightarrow c }$$ are related by $${\overrightarrow a = 8\overrightar...

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...

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow c ...

If $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \t...

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...

Let $$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1...

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar vectors and $$\lambda $$ is a real number then $$\left[ {\lambda \lef...

For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \tim...

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 $$...

If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then :

Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ be such that $$\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow ...

If $${\overrightarrow a ,\overrightarrow b ,\overrightarrow c }$$ are non-coplanar vectors and $$\lambda $$ is a real number, then the vectors $${\ove...

Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of these are collinear. If the vec...

Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be non-zero vectors such that $$\left( {\overrightarrow a \times \overrighta...

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 ...

If $$\overrightarrow u \,,\overrightarrow v $$ and $$\overrightarrow w $$ are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overr...

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...

If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr ...

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 :

The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are th...

Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat ...

If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then...

$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are $$3$$ vectors, such that $$\overrightarrow a + \overrightarrow b + \overrighta...

If $$\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c $$ are vectors such that $$\left[ {\overrightarrow a \,\overrightarrow b ...

If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respec...

If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$$ thus what will be the v...

$$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vectors and $$\overrightarrow c $$ i...

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...

If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrightarrow a }$$ and $${\overrigh...

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...

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...

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...

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...

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...

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...

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 ...

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...

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...

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, ...

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...

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>...

$$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$$ and $$D(4,5,0),|\lambda| \leq 5$$ are the vertices of a quadrilateral $$A B C D$$. If its area is 18 square ...

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...

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 $...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero non-coplanar vectors. Let the position vectors of four...

Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrigh...

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...

Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-coplanar vectors such that $$\overrightarrow a $$ $$\times$$ $...

Let  $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,   $$\overrightarrow b = \widehat i + \widehat j + \widehat k$$ &...

If $$\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k$$ and $$\overrightar...

Let $$\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$$, $$\lambda$$ $$\in$$ R. If $$\overrightarrow a $$ is a vector such that $$\o...

Let $$\theta$$ be the angle between the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$, where $$|\overrightarrow a | = 4,$$ $$|\overrightar...

Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector $...

Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$$ and $$...

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...

Let $$\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$$,   $$\overrightarrow b = 3\widehat i + \beta \widehat j - \a...

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vec...

If $$\left( {\overrightarrow a + 3\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 5\overrightarrow b } \right)$$ an...

Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors....

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...

Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three mutually perpendicular vectors of the same magnitude and equally i...

If the shortest distance between the lines $$\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\wideha...

Let $$\overrightarrow x $$ be a vector in the plane containing vectors $$\overrightarrow a = 2\widehat i - \widehat j + \widehat k$$ and $$\overright...

If $$\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$$,$$\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \wideh...

Let $$\overrightarrow c $$ be a vector perpendicular to the vectors, $$\overrightarrow a $$ = $$\widehat i$$ + $$\widehat j$$ $$-$$ $$\widehat k$$ and...

Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. I...

Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and $$\overrightarrow c = \wideh...

Let three vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow c $$ is coplanar with $$\ov...

If $$\overrightarrow x $$ and $$\overrightarrow y $$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \l...

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are unit vectors, then the greatest value of $$\sqrt 3 \left| {\overrightarrow a + \overrightarr...

Let the vectors $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be such that $$\left| {\overrightarrow a } \right| = 2$$, $$\l...

If $$\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$$, then the value of $${\left| {\widehat i \times \left( {\overrightarrow a \times ...

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...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $${\left| {\overrightarrow a - \overrig...

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\left| {\overrightarrow a } \right| = \sqrt...

If the vectors, $$\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k$$, $$\overrightarrow q = a\widehat i + \left( {a...