Dot Product of Two Vectors

Two projectiles are fired from the same point with the same speed at angles of projection $60°\mathrm{and}30°$ respectively. The correct statement is

$\overrightarrow{OA}=2\hat{i}+3\hat{j}+4\hat{k}$, $\overrightarrow{OB}=4\hat{i}+5\hat{j}+2\hat{k}$ and $\overrightarrow{BA}=-2\hat{i}-2\hat{j}+2\hat{k}$ are three sides of a triangle. Prove the triangle inequality that sum of two sides is greater than third side.

Find the projection of the vector $\hat{i}+\hat{j}+7\hat{k}$ on the vector $7\hat{i}-\hat{j}+8\hat{k}$.

Find the angle $\mathrm{\theta }$ between vectors $2\hat{i}-\hat{j}$ and $\hat{i}+2\hat{j}$.

If the direction cosines of two lines are given by $l+3\mathrm{m}+5\mathrm{n}=0$ and $5l\mathrm{m}-2\mathrm{mn}+6\ln =0$, then the angle between the lines is

It is given that $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are vectors of lengths $6, 8, 10$ respectively. If $\overrightarrow{a}$ is perpendicular to $(\overrightarrow{b}+\overrightarrow{c}), \overrightarrow{b}$ is perpendicular $(\overrightarrow{c}+\overrightarrow{a});$ and $\overrightarrow{c}$ is perpendicular to $\left(\overrightarrow{a}+\overrightarrow{b}\right),$ then the length of the vector $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ is

Let $\overrightarrow{u}=-2\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{v}=\hat{i}-2\hat{j}+2\hat{k}.$ Then the component of $v$ on $u$ is

What is the angle between the vectors $(\hat{\mathrm{i}}+\hat{\mathrm{j}})$ and $(\hat{\mathrm{j}}+\hat{\mathrm{k}})$ ?

What will be the angle between the two vectors $\overrightarrow{\mathrm{A}}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{B}}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}-5\hat{\mathrm{k}}$ ?

Evaluate : $\frac{{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2}{2{\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2}$

$\overrightarrow{P}\hspace{0.17em}\text{and}\hspace{0.17em}\overrightarrow{Q}$ are two non-zero vectors inclined to each other at an angle $\theta .$  $\hat{\mathrm{p}} \text{and} \hspace{0.17em}\hat{\mathrm{q}}$ are unit vectors along $\overrightarrow{P}\hspace{0.17em}and\hspace{0.17em}\overrightarrow{Q}$ respectively. The component of $\overrightarrow{Q}$ in the direction of $\overrightarrow{P}$  will be

If the projection of $\overrightarrow{PQ}$ on the axes $OX,OY,OZ$ are respectively $12,3$ and $4,$ then the magnitude of $\overrightarrow{PQ}$ is

The component of a vector $r$ along $X$-axis will have a maximum value, if

If $\overrightarrow{A}=2\widehat{i}+3\widehat{j}+8\widehat{k}$ is perpendicular to $\overrightarrow{B}=4\widehat{j}-4\widehat{i}+\alpha \widehat{k}$ , then the value of $\alpha$ is

The vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are such that $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\left|\overrightarrow{A}-\overrightarrow{B}\right|.$ The angle between the two vectors will be