JEE Main
Mathematics
Vector Algebra
Previous Years Questions

## MCQ (Single Correct Answer)

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 $$\overr... Let : \vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k} and \vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k} be three vectors. If \vec{... Let$$\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 $$\ove... If$$\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}=...
$$\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b... A vector$$\vec{a}$$is parallel to the line of intersection of the plane determined by the vectors$$\hat{i}, \hat{i}+\hat{j}$$and the plane determi... Let$$\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...
Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$, $$\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$$ and $$\overrightar... If$$\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 $$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\wide... Let$$\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k$$and$$\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k$$be two vector... If the volume of parallelopiped formed by the vectors$$\widehat i + \lambda \widehat j + \widehat k$$,$$\widehat j + \lambda \widehat k$$and$$\lam...
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 $$\mathop b\limits^ \to ... The magnitude of the projection of the vector$$\mathop {2i}\limits^ \wedge + \mathop {3j}\limits^ \wedge + \mathop k\limits^ \wedge $$on the ve... Let$$\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 $$\overrightarrow b$$ two unit vectors. If the vectors $$\,\overrightarrow c = \widehat a + 2\widehat b$$ and $$\over... Let$$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 $$\overrightarrow a = \widehat j - \widehat k$$ 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\,\,\...
The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vector are:
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 $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \t... The values of a, for which points$$A, B, C$$with position vectors$$2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat ...
If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then
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 $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ 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 $$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1... A particle acted on by constant forces$$4\widehat i + \widehat j - 3\widehat k$$and$$3\widehat i + \widehat j - \widehat k$$is displaced from the ... Let$$\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 $$be non-zero vectors such that$$\left( {\overrightarrow a \times \overrighta... 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 $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat ... The vectors$$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$are th...$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$are$$3$$vectors, such that$$\overrightarrow a + \overrightarrow b + \overrighta... 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 If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr ... Consider points$$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 $$\overrightarrow u \,,\overrightarrow v$$ and $$\overrightarrow w$$ are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overr... If$$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$and the angle between$${\overrightarrow a }$$and$${\overrigh... If $$\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c$$ are vectors such that $$\left[ {\overrightarrow a \,\overrightarrow b ... If$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$are vectors show that$$\overrightarrow a + \overrightarrow b + \overrightarrow ... 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 $$\ov... If$$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$then... ## Numerical Let$$\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>... 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 $$|\overrightarrow a | = 4,$$ $$|\overrightar... If the shortest distance between the lines$$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\wideha...
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...
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