Vectors Questions with Solutions

If $\hat{a}=\hat{i}-\hat{k}$, $\hat{b}=x\hat{i}+\hat{j}+\left(1-x\right)\hat{k}$ and $\hat{c}=y\hat{i}+x\hat{j}+\left(1+x-y\right)\hat{k}$, then $\left[\hat{a}\hat{b}\hat{c}\right]$ depends on

If $\hat{a}$ and $\hat{b}$ are unit vectors such that $\left|2\hat{a}+\hat{b}\right|=3$, then which of the following statements is true?

Let $\hat{A}=2\hat{i}+\hat{j}-2\hat{k}$ and $\hat{B}=\hat{i}+\hat{j}$. If $\hat{C}$ is a vector such that $\left|\hat{C}-\hat{A}\right|=3$, the angle between $\hat{A}\times \hat{B}$ and $\hat{C}$ is ${30}^{\circ }$, and $\left|\left(\hat{A}\times \hat{B}\right)\times \hat{C}\right|=3$, then the value of $\hat{A}\cdot \hat{C}$ is

Let $\hat{a}=\hat{j}-\hat{k}$ and $\hat{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\hat{b}$ satisfying $\left(\hat{a}\times \hat{b}\right)+\hat{c}=\hat{0}$ and $\hat{a}\cdot \hat{b}=3$ is

The area of the parallelogram whose diagonals are $\hat{a}=3\hat{i}+\hat{j}-2\hat{k}$ and $\hat{b}=\hat{i}-3\hat{j}+4\hat{k}$ is

Let $\hat{a}$, $\hat{b}$, $\hat{c}$ be the position vectors of three vertices A, B, C of a triangle respectively. Then the area of this triangle is given by

Let $\hat{a}=x\hat{i}-3\hat{j}-\hat{k}$ and $\hat{b}=2x\hat{i}+x\hat{j}-\hat{k}$. The angle between $\hat{a}$ and $\hat{b}$ is acute, and the angle between $\hat{b}$ and the positive $y$ -axis lies in $\left(\frac{\pi }{2},\pi \right)$. The set of all such values of $x$ is

If $\hat{v}=2\hat{i}+\hat{j}-\hat{k}$, $\hat{w}=\hat{i}+3\hat{k}$ and $\hat{u}$ is a unit vector, then the maximum value of the scalar triple product $\left[\hat{u}\hat{v}\hat{w}\right]$ is

A vector $\hat{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. The system is rotated through a certain angle about the origin, and with respect to the new system $\hat{a}$ has components $p+1$ and $1$. Then

Let $\hat{a},\hat{b},\hat{c}$ be three non-zero vectors, no two of which are collinear. If $\hat{a}+\hat{b}$ is collinear with $\hat{c}$ and $\hat{b}+\hat{c}$ is collinear with $\hat{a}$, then $\hat{a}+\hat{b}+\hat{c}$ is equal to

If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are unit vectors, then ${\left|\hat{a}-\hat{b}\right|}^2+{\left|\hat{b}-\hat{c}\right|}^2+{\left|\hat{c}-\hat{a}\right|}^2$ does not exceed

If $\hat{a},\hat{b},\hat{c}$ are non-coplanar unit vectors such that $\hat{a}\times \left(\hat{b}\times \hat{c}\right)=\frac{\hat{b}+\hat{c}}{\sqrt{2}}$, then the angle between $\hat{a}$ and $\hat{b}$ is:

The value of $\lambda$ for which the volume of the parallelepiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}$, $\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum is

The value of $\lambda$ such that the four points whose position vectors are $3\hat{i}-2\hat{j}+\lambda \hat{k}$, $6\hat{i}+3\hat{j}+\hat{k}$, $5\hat{i}+7\hat{j}+3\hat{k}$ and $2\hat{i}+2\hat{j}+6\hat{k}$ are coplanar is

Let $\hat{A}=2\hat{i}+\hat{j}-2\hat{k}$ and $\hat{B}=\hat{i}+\hat{j}$. If $\hat{C}$ is a vector such that $\hat{A}\cdot \hat{C}=\left|\hat{C}\right|$, $\left|\hat{C}-\hat{A}\right|=2\sqrt{2}$ and the angle between $\hat{A}\times \hat{B}$ and $\hat{C}$ is ${30}^{\circ }$, then $\left|\left(\hat{A}\times \hat{B}\right)\times \hat{C}\right|$ is equal to

A rigid body is rotating at the rate of $3$ radians per second about an axis $AB$, where $A\left(1,-2,1\right)$ and $B\left(3,-4,2\right)$. The velocity of the point $P\left(5,-1,-1\right)$ of the body is

Forces of magnitude 5, 3, 1 units act in the directions $6\hat{i}+2\hat{j}+3\hat{k}$, $3\hat{i}-2\hat{j}+6\hat{k}$, $2\hat{i}-3\hat{j}-6\hat{k}$ respectively on a particle which is displaced from the point (2, −1, −3) to (5, −1, 1). The total work done by the force is

If $\hat{a},\hat{b},\hat{c}$ are three non-zero vectors with no two of which are collinear, $\hat{a}+2\hat{b}$ is collinear with $\hat{c}$ and $\hat{b}+3\hat{c}$ is collinear with $\hat{a}$, then $\left|\hat{a}+2\hat{b}+6\hat{c}\right|$ will be equal to

The area of the triangle whose vertices have position vectors $3\hat{i}+\hat{j}$, $5\hat{i}+2\hat{j}+\hat{k}$ and $\hat{i}-2\hat{j}+3\hat{k}$ is

If the vectors $a\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ $\left(a,b,c\ne 1\right)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$