Vector Algebra · JEE Main previous year questions

The sum $\frac{3\times 1^3}{1^2}+\frac{5\times (1^3+2^3)}{1^2+2^2}+\frac{7\times (1^3+2^3+3^3)}{1^2+2^2+3^2}+\dots$ upto 10 terms is:

1. A.600
2. B.660
3. C.680
4. D.620

The distance of the point having position vector $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line passing through the point (2, 3, - 4) and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is :

1. A.6
2. B.7
3. C.$2\sqrt{13}$
4. D.$4\sqrt{3}$

Let $\vec{\alpha }=3\hat{i}+\hat{j}$ and $\vec{\beta }=2\hat{i}-\hat{j}+3\hat{k}$. If $\vec{\beta }={\vec{\beta }}_1-{\vec{\beta }}_2$, where ${\vec{\beta }}_1$ is parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ is perpendicular to $\vec{\alpha }$, then ${\vec{\beta }}_1\times {\vec{\beta }}_2$ is equal to

1. A.$3\hat{i}-9\hat{j}-5\hat{k}$
2. B.$\frac{1}{2}(-3\hat{i}+9\hat{j}+5\hat{k})$
3. C.$-3\hat{i}+9\hat{j}+5\hat{k}$
4. D.$\frac{1}{2}(3\hat{i}-9\hat{j}+5\hat{k})$

If a unit vector $\vec{a}$ makes angles $\pi /3$ with $\hat{i}$, $\pi /4$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k}$, then a value of $\theta$ is :-

1. A.$\frac{5\pi }{6}$
2. B.$\frac{5\pi }{12}$
3. C.$\frac{2\pi }{3}$
4. D.$\frac{\pi }{4}$

Let $\vec{a}=3\hat{i}+2\hat{j}+x\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$, for some real x. Then $|\vec{a}\times \vec{b}|=r$ is possible if :

1. A.$0<r<\sqrt{\frac{3}{2}}$
2. B.$3\sqrt{\frac{3}{2}}<r<5\sqrt{\frac{3}{2}}$
3. C.$r\ge 5\sqrt{\frac{3}{2}}$
4. D.$\sqrt{\frac{3}{2}}<r\le 3\sqrt{\frac{3}{2}}$

The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0 is :

1. A.$\vec{r}\times (\hat{i}-\hat{k})-2=0$
2. B.$\vec{r}\cdot (\hat{i}+\hat{k})+2=0$
3. C.$\vec{r}\cdot (\hat{i}-\hat{k})+2=0$
4. D.$\vec{r}\times (\hat{i}-\hat{k})+2=0$

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k}$, $\hat{i}+\mu \hat{j}+\hat{k}$, $\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is :

1. A.2
2. B.$-1$
3. C.0
4. D.1

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{1}{2}\vec{b}$, then $|\alpha -\beta |$ is equal to :

1. A.$90^{\circ }$
2. B.$30^{\circ }$
3. C.$45^{\circ }$
4. D.$60^{\circ }$

Let $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $\beta \hat{i}+(1-\beta )\hat{j}$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :

1. A.4
2. B.1
3. C.2
4. D.3

Let $\vec{a}=\hat{i}-\hat{j}$, $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times \vec{c}+\vec{b}=\vec{0}$ and $\vec{a}\cdot \vec{c}=4$, then $|\vec{c}{|}^2$ is equal to :

1. A.8
2. B.$\frac{19}{2}$
3. C.9
4. D.$\frac{17}{2}$

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{c}=\hat{j}-\hat{k}$ and a vector $\vec{b}$ be such that $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{a}\cdot \vec{b}=3$. Then $|\vec{b}|$ equals :

1. A.$\frac{11}{3}$
2. B.$\frac{11}{\sqrt{3}}$
3. C.$\sqrt{\frac{11}{3}}$
4. D.$\frac{\sqrt{11}}{3}$

If $\lambda \in \mathbf{R}$ is such that the sum of the cubes of the roots of the equation, $x^2+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

1. A.$4\sqrt{2}$
2. B.$2\sqrt{5}$
3. C.$2\sqrt{7}$
4. D.20

If $\vec{a},\vec{b}$, and $\vec{c}$ are unit vectors such that $\vec{a}+2\vec{b}+2\vec{c}=\vec{0}$, then $|\vec{a}\times \vec{c}|$ is equal to :

1. A.$\frac{\sqrt{15}}{4}$
2. B.$\frac{1}{4}$
3. C.$\frac{15}{16}$
4. D.$\frac{\sqrt{15}}{16}$

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$. If $\vec{u}$ is perpendicular to $\vec{a}$ and $\vec{u}\cdot \vec{b}=24$, then $|\vec{u}{|}^2$ is equal to

1. A.$336$
2. B.$315$
3. C.$256$
4. D.$84$

The sum of all the real values of x satisfying the equation $2^{(x-1)(x^2+5x-50)}=1$ is :

1. A.$16$
2. B.$14$
3. C.$-4$
4. D.$-5$

If the vector $\vec{b}=3\hat{j}+4\hat{k}$ is written as the sum of a vector ${\vec{b}}_1$, parallel to $\vec{a}=\hat{i}+\hat{j}$ and a vector ${\vec{b}}_2$, perpendicular to $\vec{a}$, then ${\vec{b}}_1\times {\vec{b}}_2$ is equal to :

1. A.$-3\hat{i}+3\hat{j}-9\hat{k}$
2. B.$6\hat{i}-6\hat{j}+\frac{9}{2}\hat{k}$
3. C.$-6\hat{i}+6\hat{j}-\frac{9}{2}\hat{k}$
4. D.$3\hat{i}-3\hat{j}+9\hat{k}$

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i}-6\hat{j}$ and $3\hat{i}+4\hat{j}-12\hat{k}$, is :

1. A.$26$
2. B.$65$
3. C.$20$
4. D.$52$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}\times \left(\vec{b}\times \vec{c}\right)=\frac{\sqrt{3}}{2}\left(\vec{b}+\vec{c}\right)$. If $\vec{b}$ is not parallel to $\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

1. A.$\frac{2\pi }{3}$
2. B.$\frac{5\pi }{6}$
3. C.$\frac{3\pi }{4}$
4. D.$\frac{\pi }{2}$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $\left(\vec{a}\times \vec{b}\right)\times \vec{c}=\frac{1}{3}\left|\vec{b}\right|\left|\vec{c}\right|\vec{a}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$, then a value of $\sin \theta$ is :

1. A.$\frac{2}{3}$
2. B.$\frac{-2\sqrt{3}}{3}$
3. C.$\frac{2\sqrt{2}}{3}$
4. D.$\frac{-\sqrt{2}}{3}$

The distance of the point $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line that passes through the point $2\hat{i}+3\hat{j}-4\hat{k}$ and is parallel to the vector $6\hat{i}+3\hat{j}-4\hat{k}$ is

1. A.9
2. B.8
3. C.7
4. D.10

Statement 1: The vectors $\vec{a},\vec{b}$ and $\vec{c}$ lie in the same plane if and only if $\vec{a}\cdot (\vec{b}\times \vec{c})=0$ Statement 2: The vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u}\cdot \vec{v}=0$ where $\vec{u}\times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$

1. A.Statement 1 is false, Statement 2 is true.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation for Statement 1.
3. C.Statement 1 is true, Statement 2 is false.
4. D.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

If $\vec{u}=\hat{j}+4\hat{k},\vec{v}=\hat{i}+3\hat{k}$ and $\vec{w}=\cos \theta \hat{i}+\sin \theta \hat{j}$ are vectors in 3-dimensional space, then the maximum possible value of $|\vec{u}\times \vec{v}\cdot \vec{w}|$ is

1. A.$\sqrt{3}$
2. B.5
3. C.$\sqrt{14}$
4. D.7

A unit vector which is perpendicular to the vector $2\hat{i}-\hat{j}+2\hat{k}$ and is coplanar with the vectors $\hat{i}+\hat{j}-\hat{k}$ and $2\hat{i}+2\hat{j}-\hat{k}$ is

1. A.$\frac{2\hat{j}+\hat{k}}{\sqrt{5}}$
2. B.$\frac{3\hat{i}+2\hat{j}-2\hat{k}}{\sqrt{17}}$
3. C.$\frac{3\hat{i}+2\hat{j}+2\hat{k}}{\sqrt{17}}$
4. D.$\frac{2\hat{i}+2\hat{j}-\hat{k}}{3}$

$ABCD$ is parallelogram. The position vectors of $A$ and $C$ are respectively, $3\hat{i}+3\hat{j}+5\hat{k}$ and $\hat{i}-5\hat{j}-5\hat{k}$. If $M$ is the midpoint of the diagonal $DB$, then the magnitude of the projection of $\overrightarrow{OM}$ on $\overrightarrow{OC}$, where $O$ is the origin, is

1. A.$7\sqrt{51}$
2. B.$\frac{7}{\sqrt{50}}$
3. C.$7\sqrt{50}$
4. D.$\frac{7}{\sqrt{51}}$

If $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+(2\lambda -1)\hat{k}$ are coplanar vectors, then $\lambda$ is equal to

1. A.0
2. B.$-1$
3. C.2
4. D.1