Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: Exercise

Show that three points $A(-2\hat{i}+3\hat{j}+5\hat{k}), B(\hat{i}+2\hat{j}+3\hat{k}) \mathrm{and} C(7\hat{i}-\hat{k})$ are collinear.

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

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 $7:1$ internally 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

If $\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k}$. Find magnitude of $6\overrightarrow{a}$.

$\overrightarrow{b}=(2,3,5)\text{ and }\overrightarrow{a}=(2,1,3)$ are vectors in Cartesian form. Find $\overrightarrow{a}+\overrightarrow{b}$.

$\overrightarrow{b}=\left(1,2,3\right)$ and $\overrightarrow{a}=\left(2,4,6\right)$ are vectors in Cartesian form. Find $\overrightarrow{a}-\overrightarrow{b}$.

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