Types of Vectors

If $\theta$ is the angle between any two vectors $\overrightarrow{a}$ and $\overrightarrow{b},$ then $|\overrightarrow{a}·\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$ then $\theta$ is equal to

The value of $\hat{i}·(\hat{j}\times \hat{k})+\hat{j}·(\hat{i}\times \hat{k})+\hat{k}·(\hat{i}\times \hat{j})$ is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\overrightarrow{a}+\overrightarrow{b}$ is a unit vector if

If $\theta$ is the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b},$ then $\overrightarrow{a}·\overrightarrow{b}\ge 0$ only when

Prove that, $(\overrightarrow{a}+\overrightarrow{b})·(\overrightarrow{a}+\overrightarrow{b})=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2$ if and only if $\overrightarrow{a},\overrightarrow{b}$ are perpendicular, given $\overrightarrow{a}\ne \overrightarrow{0},\overrightarrow{b}\ne \overrightarrow{0}$.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ is equally inclined to $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$.

The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along with the sum of vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $\lambda \hat{i}+2\hat{j}+3\hat{k}$ is equal to one. Find the value of $\lambda$.

Let $\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k}$ and $\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}$. Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{c}.\overrightarrow{d}=15$.

Show that the direction cosines of a vector equally inclined to the axes $OX, OY$ and $OZ$ are $\frac{1}{\sqrt{3}}$$,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$.

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}$ and $\hat{i}-2\hat{j}-3\hat{k}$ . Find the unit vector parallel to its diagonal. And, if the area of the parallelogram is equal to $k\sqrt{5}$ sq. units. Find the value of $k$. 

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $(2\overrightarrow{a}+\overrightarrow{b})$ and $(\overrightarrow{a}-3\overrightarrow{b})$ externally in the ratio $1: 2$. Also, show that $P$ is the midpoint of the line segment $RQ$.

Show that the points $\mathrm{A}(1,-2,-8),\mathrm{B}(5,0,-2)$ and $\mathrm{C}(11,3,7)$ are collinear, and find the ratio in which $B$ divides $AC$.

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k},$ find a unit vector parallel to the vector $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}$.

Find a vector of magnitude $5$ units, and parallel to the resultant of the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k}$.

Find the value of $m+n$ for which $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector and $x=\pm \frac{m}{\sqrt{n}}$.

If $\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c},$ then is it true that $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|+\left|\overrightarrow{c}\right|?$ Justify your answer.

A girl walks $4\mathrm{km}$ towards west, then she walks $3\mathrm{km}$ in a direction $30°$ east of the north and stops. The girl's displacement from her initial point of departure is $\frac{a}{2}\hat{i}+\frac{b\sqrt{3}}{2}\hat{j}$. Then, find the value of $a+b$.

Find the scalar components and magnitude of the vector joining the points $\mathrm{P}\left(x_1,y_1,z_1\right)$ and $\mathrm{Q}\left(x_2,y_2,z_2\right)$.

Write down a unit vector in $\mathrm{XY}$ -plane, making an angle of $30°$ with the positive direction of $\mathrm{x}-$axis.