Amit M Agarwal Solutions for Chapter: Vector Algebra, Exercise 2: Exercise for Session 2

Show that the points $A, B$ and $C$ with position vectors $\overrightarrow{\mathrm{a}}=3\hat{\mathrm{i}}-4\hat{\mathrm{j}}-4\hat{\mathrm{k}},\mathrm{ }\overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\mathrm{ }$and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-3\hat{\mathrm{j}}-5\hat{\mathrm{k}}$ respectively, form the vertices of a right angled triangle.

If $\overrightarrow{a}=2\hat{i}+2\hat{j}-\hat{k}$ and $\left|x\overrightarrow{a}\right|=1$, then find the value of $x$.

If $\overrightarrow{p}=7\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{q}=3\hat{i}+\hat{j}+5\hat{k}$, then find the magnitude of $\overrightarrow{p} –2\overrightarrow{q}$.

Find a vector in the direction of  $5\hat{i}-\hat{j}+2\hat{k}$, which has magnitude $8 \mathrm{units}$.

If $\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=3\hat{i}+6\hat{j}+2\hat{k}$, then find a vector in the direction of $\overrightarrow{a}$ and having a magnitude of $\left|\overrightarrow{b}\right|$.

Find the position vector of a point $P$ which divides the line joining two points $A$ and $B$ whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$  respectively in the ratio $2 : 1$

$\left(i\right)$ internally

Find the position vector of a point $P$ which divides the line joining two points $A \& B$, whose position vectors are $\hat{i}+2\hat{j}-\hat{k} \text{and} -\hat{i}+\hat{j}+\hat{k}$, respectively in the ratio of $2:1$ externally.

If the position vector of one end of a line segment $\overrightarrow{AB}$ is $2\hat{i}+3\hat{j}-\hat{k}$ and the position vector of its middle point is $3\left(\hat{i}+\hat{j}+\hat{k}\right)$, then find the position vector of the other end.