Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 2: Exercise 2

The position vector of $\overrightarrow{A}$ and $\overrightarrow{B}$ are $2\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}}+4\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ . The length of the internal bisector of $\angle BOA$ of triangle $AOB$ is

A vector of magnitude $3$, bisecting the angle between the vectors $\overrightarrow{a}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k}$ and making an obtuse angle with $\overrightarrow{b}$ is

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar vectors and $\overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}$ are the reciprocal vectors, then $\left(l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}\right)·\left(l\overrightarrow{p}+m\overrightarrow{q}+n\overrightarrow{r}\right)$ is equal to

The resultant vector of $\overrightarrow{\mathit{P}}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$. On reversing the direction of $\overrightarrow{Q}$ the resultant vector becomes $\overrightarrow{S}$. Find the value of ${\left|\overrightarrow{R}\right|}^2+{\left|\overrightarrow{S}\right|}^2$

If the vertices of any tetrahedron be $\overrightarrow{a}=\hat{j}+2\hat{k}, \overrightarrow{b}=3\hat{i}+\hat{k}, \overrightarrow{c}=4\hat{i}+3\hat{j}+6\hat{k}$ and $\overrightarrow{d}=2\hat{i}+3\hat{j}+2\hat{k}$ then find its volume

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar non-zero vectors and $\overrightarrow{\mathit{r}}$ is any vector in space, then $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \left(\overrightarrow{r}\times \overrightarrow{c}\right)+\left(\overrightarrow{b}\times \overrightarrow{c}\right)\times \left(\overrightarrow{r}\times \overrightarrow{a}\right)+\left(\overrightarrow{c}\times \overrightarrow{a}\right)\times \left(\overrightarrow{r}\times \overrightarrow{b}\right)$ is equal to

The moment about the point $\hat{i}+2\hat{j}+3\hat{k}$ of a force represented by $\hat{i}+\hat{j}+\hat{k}$ acting through the point $2\hat{i}+3\hat{j}+\hat{k}$ is 

Let the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be perpendicular to each other and the unit vector $\overrightarrow{c}$ be inclined at an angle $\theta$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+z\left(\overrightarrow{a}\times \overrightarrow{b}\right)$, then