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?

A vector of magnitude 5 units makes equal angles with each of the three mutually perpendicular coordinate axes. The sum of the magnitudes of its projections on the three axes is

If $\hat{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\hat{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$, then the value of $\left(2\hat{a}+\hat{b}\right)\cdot \left[\left(\hat{a}\times \hat{b}\right)\times \left(\hat{a}-2\hat{b}\right)\right]$ 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 $\left|\hat{C}-\hat{A}\right|=3$, 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

If the vectors $2\hat{i}-3\hat{j}$, $\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{k}$ form three conterminous edges of a parallelepiped, then the volume of the parallelepiped is

Forces $3\hat{i}+2\hat{j}+5\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ act on a particle and displace it from the point $2\hat{i}-\hat{j}-3\hat{k}$ to the point $4\hat{i}-3\hat{j}+7\hat{k}$. Then the work done by the forces 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

If the three vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda \hat{j}+5\hat{k}$ are coplanar, then $\lambda$ 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

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are unit vectors such that $\hat{a}+\hat{b}+\hat{c}=\hat{0}$, then the value of $\hat{a}\cdot \hat{b}+\hat{b}\cdot \hat{c}+\hat{c}\cdot \hat{a}$ is

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are coplanar vectors, then the value of $\left[2\hat{a}-\hat{b}2\hat{b}-\hat{c}2\hat{c}-\hat{a}\right]$ is

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

If $\left|\hat{a}\times \hat{b}\right|=\left|\hat{a}\cdot \hat{b}\right|$, then the angle between $\hat{a}$ and $\hat{b}$ is

$ABCD$ is a parallelogram with diagonals $AC$ and $BD$. Then $\widehat{AC}-\widehat{BD}$ is equal to

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

The vectors $\hat{a},\hat{b},\hat{c}$ are equal in length and, taken pairwise, make equal angles. If $\hat{a}=\hat{i}+\hat{j}$, $\hat{b}=\hat{j}+\hat{k}$ and $\hat{c}$ makes an obtuse angle with $\hat{i}$, then $\hat{c}$ is equal to

The position vectors of the points $A,B,C,D$ are $\hat{i}+\hat{j}+\hat{k}$, $2\hat{i}+5\hat{j}$, $3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ respectively. The angle between $\widehat{AB}$ and $\widehat{CD}$ is