Vectors Questions with Solutions

Let $\hat{a}=\hat{i}+\hat{j}$ and $\hat{b}=2\hat{i}-\hat{k}$. The point of intersection of the lines $\hat{r}\times \hat{a}=\hat{b}\times \hat{a}$ and $\hat{r}\times \hat{b}=\hat{a}\times \hat{b}$ is

The angle between $\hat{a}$ and $\hat{b}$ is ${120}^{\circ }$. If $\left|\hat{b}\right|=2\left|\hat{a}\right|$ and the vectors $\hat{a}+x\hat{b}$ and $\hat{a}-\hat{b}$ are at right angles, then $x=$

Let $a,b,c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is

If the volume of the parallelepiped whose adjacent edges are $\hat{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\hat{b}=\hat{i}+\alpha \hat{j}+2\hat{k}$ and $\hat{c}=\hat{i}+2\hat{j}+\alpha \hat{k}$ is 15, then $\alpha$ is equal to

Let $\hat{a}=2\hat{i}+2\hat{j}+\hat{k}$ and $\hat{b}$ be another vector such that $\hat{a}\cdot \hat{b}=14$ and $\hat{a}\times \hat{b}=3\hat{i}+\hat{j}-8\hat{k}$. Then the vector $\hat{b}$ is

If $\hat{a}=\lambda \hat{i}+\hat{j}-2\hat{k}$, $\hat{b}=\hat{i}+\lambda \hat{j}-2\hat{k}$, $\hat{c}=\hat{i}+\hat{j}+\hat{k}$ and $\left[\hat{a}\hat{b}\hat{c}\right]=7$, then the values of $\lambda$ are

If $\left(\hat{a}\times \hat{b}\right)\times \hat{c}=\hat{a}\times \left(\hat{b}\times \hat{c}\right)$, then

If $\hat{A}=4\hat{i}+3\hat{j}+\hat{k}$ and $\hat{B}=2\hat{i}-\hat{j}+2\hat{k}$, then the unit vector $\hat{N}$ perpendicular to the vectors $\hat{A}$ and $\hat{B}$, such that $\hat{A}$, $\hat{B}$ and $\hat{N}$ form a right handed system, is:

If $\widehat{AC}=2\hat{i}+\hat{j}+\hat{k}$ and $\widehat{BD}=-\hat{i}+3\hat{j}+2\hat{k}$, then the area of the quadrilateral ABCD is

If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are vectors such that $\hat{a}+\hat{b}+\hat{c}=\hat{0}$ and $\left|\hat{a}\right|=7$, $\left|\hat{b}\right|=5$, $\left|\hat{c}\right|=3$, then the angle between the vectors $\hat{b}$ and $\hat{c}$ is

The vector $\hat{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\hat{b}=\hat{i}+\hat{j}$ and $\hat{c}=\hat{j}+\hat{k}$ and bisects the angle between $\hat{b}$ and $\hat{c}$. Then which of the following gives possible values of $\alpha$ and $\beta$?

A bird is flying in a straight line with velocity vector $10\hat{i}+6\hat{j}+\hat{k}$, measured in km/hr. If the starting point is $\left(1,2,3\right)$, how much time does it take to reach a point in space that is 13 m high from the ground?

The position vectors of the vertices A, B, C of a tetrahedron ABCD are $\hat{i}+\hat{j}+\hat{k}$, $\hat{i}$ and $3\hat{i}$ respectively, and the altitude from the vertex D to the opposite face ABC meets the face at E. If the volume of the tetrahedron is $\frac{2\sqrt{2}}{3}$, then the length of DE is:

$\hat{a}$ and $\hat{b}$ are non-zero, non-collinear vectors such that $\left|\hat{a}\right|=2$, $\hat{a}\cdot \hat{b}=1$ and the angle between $\hat{a}$ and $\hat{b}$ is $\frac{\pi }{3}$. If $\hat{r}$ is any vector satisfying $\hat{r}\cdot \hat{a}=2$, $\hat{r}\cdot \hat{b}=8$, $\left(\hat{r}+2\hat{a}-10\hat{b}\right)\cdot \left(\hat{a}\times \hat{b}\right)=6$ and $\hat{r}+2\hat{a}-10\hat{b}=\lambda \left(\hat{a}\times \hat{b}\right)$, then $\lambda =$

The value of $m$ for which the volume of the parallelepiped is 4 cubic units, whose three edges are represented by $\hat{a}=m\hat{i}+\hat{j}+\hat{k}$, $\hat{b}=\hat{i}-\hat{j}+\hat{k}$ and $\hat{c}=\hat{i}+2\hat{j}-\hat{k}$, is

The number of distinct real values of $\lambda$ for which the vectors ${\lambda }^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+{\lambda }^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+{\lambda }^2\hat{k}$ are coplanar is

The vector $\hat{A}=\left(2x+1\right)\hat{i}+\left(x^2-6y\right)\hat{j}+\left(x{y}^2+3z\right)\hat{k}$ is a

Let $\hat{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\hat{b}=\hat{i}+2\hat{j}-\hat{k}$ and $\hat{c}=3\hat{i}+\hat{j}+\lambda \hat{k}$ be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of $\lambda$ is