Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: CBSE-2019

If $\theta$ is the angle between two vectors $\hat{i}-2\hat{j}+3\hat{k}\text{ and }3\hat{i}-2\hat{j}+\hat{k}$, find $\sin \theta$.

Let $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}$, and $\overrightarrow{\mathrm{c}}=3\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$. Find a $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$, and $\overrightarrow{b}$ and $\overrightarrow{\mathrm{d}}·\overrightarrow{\mathrm{a}}=21$

If $\overrightarrow{a}=2\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{c}=-3\hat{i}+\hat{j}+2\hat{k}$  find $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$.

If the projection of $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k}\text{ on }\overrightarrow{b}=2\hat{i}+\lambda \hat{k}$ is zero, then the value of $\lambda$ is 

The position vectors of two points $A$ and $B$ are $\overline{\mathrm{OA}}=2\hat{i}-\hat{j}-\hat{k}$ and $\overline{\mathrm{OB}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}$, respectively. The position vector of a point $P$ which divides the line segment joining $A$ and $B$ in the ratio $2:1$ is

Find a unit vector perpendicular to each of the vectors $\overrightarrow{a}\text{ and }\overrightarrow{b}$ where $\overrightarrow{a}=5\hat{i}+6\hat{j}-2\hat{k}\text{ and }\overrightarrow{b}=7\hat{i}+6\hat{j}+2\hat{k}$.

Find the volume of the parallelepiped whose adjacent edges are represented by $2\overrightarrow{a},-\overrightarrow{b}\text{ and }3\overrightarrow{c}$, where $\overrightarrow{a}=\hat{i}-\hat{j}+2\hat{k},\overrightarrow{b}=3\hat{i}+4\hat{j}-5\hat{k},\text{ and } \overrightarrow{c}=2\hat{i}-\hat{j}+3\hat{k}$.

If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is $\sqrt{3}$.