Scalars and Vectors

The position vectors of two points $A$ and $B$ are $\hat{i}-\hat{j}$ and $\hat{j}+\hat{k}$ respectively.

Consider the following points :

$1.$ $\left(-1,-3, 1\right)$

$2.$ $\left(-1, 3, 2\right)$

$3.$ $\left(-2, 5, 3\right)$

What is the magnitude of $\overrightarrow{AB}$?

The position vectors of the vertices $A,B$ and $C$ of a triangle are $\hat{i}+\hat{j}, \hat{j}+\hat{k}$ and $\hat{i}+\hat{k}$ respectively. Find a unit vector lying in the plane of $ABC$ and perpendicular to $IA,$ where $I$ is the incentre of the triangle.

$\overrightarrow{A}=-\sqrt{3} \hat{i}+\hat{j}$ is rotated ${90}^{\circ }$ clockwise. Find the new vector.

For any vector $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, with $10\left|a_i\right|<1,i=1,2,3$, consider the following statements:
$\left(A\right)$ : $\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}\le \left|\overrightarrow{a}\right|$
$\left(B\right)$ : $\left|\overrightarrow{a}\right|\le 3\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}$

If unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ make an angle $\frac{\mathrm{\pi }}{6}$ and $\frac{2\mathrm{\pi }}{3}$ with $x-$axis respectively, then $\left|\overrightarrow{a}+\overrightarrow{b}\right|=?$

Slect which of the above vectors have the same direction.

Given that $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=0$, out of three vectors two are equal in magnitude and the magnitude of third vector is $\sqrt{2}$ times that of either of two having equal magnitudes. Then, angle between vectors are given by

 $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are the position vectors of the points $A=(x,y,z) B=(y,-2z,3x) C=(2z,3x,-y)$ and $D=(1,-1,2)$ respectively. If $\left|\overrightarrow{a}\right|=2\sqrt{3},(\overrightarrow{a}^\overrightarrow{b})=(\overrightarrow{a}^\overrightarrow{c}), (\overrightarrow{a}^\overrightarrow{d})=\frac{\mathrm{\pi }}{2}$ and $(\overrightarrow{a}^\overrightarrow{j})$is obtuse, then find x, y, z

Find the magnitude of the vector $3\hat{i}-12\hat{j}+3\hat{k}$.

If $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=4,\left|\overrightarrow{c}\right|=2$ then minimum value of $\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|$ is 

A particle is acted upon by constant forces ${\overrightarrow{F}}_1= \left(2\hat{i}-3\hat{j}+4\hat{k}\right)$ and ${\overrightarrow{F}}_2= \left(-\hat{i}+2\hat{j}-3\hat{k}\right)$ and is displaced from the point $A\left(2,1,0\right)$ to the point $B\left(-3,-4,2\right)$. Find the total work done by the forces.

Explain vectors.

Represent graphically a displacement of $40 \mathrm{km}, 30°$ west of south.

Find a vector whose magnitude is $5$ and whose direction is the same as of $\overrightarrow{b}=2\hat{i}+3\hat{j}+2\sqrt{3}\hat{k}$.

A particle is moving at $5 \mathrm{m} {\mathrm{s}}^{-1}$ towards east. In one second its velocity changes to $5 \mathrm{m} {\mathrm{s}}^{-1}$ towards north. Assuming the acceleration to be uniform, the change in velocity will be directed at

Which of the following is a scalar quantity?

Which is the only vector quantity out of the following?

Surface area is,

Volume is