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

Let a vector $\mathrm{\alpha }\hat{\mathrm{i}}+\mathrm{\beta }\hat{\mathrm{j}}$ be obtained by rotating the vector $\sqrt{3}\hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $45°$ about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $\left(\alpha ,\beta \right),\left(0,\beta \right)$ and $\left(0,0\right)$ is equal to

Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k}, x, y\in R, x>0,$ be such that $\left|\overrightarrow{PQ}\right|=\sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OQ}.$ If $\overrightarrow{OR}=3\hat{i}+\mathrm{z}\hat{j}-7\hat{k}, z\in R,$ is coplanar with $\overrightarrow{OP}$ and $\overrightarrow{OQ},$ then the value of $x^2+y^2+z^2$ is equal to

Let $\overrightarrow{\alpha }=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\beta }=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ and $\overrightarrow{\gamma }=2\hat{\mathrm{i}}+\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ be three vectors. If $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are both perpendicular to the vector $\overrightarrow{\delta }$ and $\overrightarrow{\delta }·\overrightarrow{\gamma }=10$, then what is the magnitude of $\overrightarrow{\delta }$ ?

Let $\overrightarrow{p}$ and $\overrightarrow{q}$ be the position vectors of the points $P$ and $Q$ respectively, with respect to origin $O.$ The points $R$ and $S$ divide $PQ$ internally and externally respectively, in the ratio $2: 3.$ If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular, then which one of the following is correct?

If $\left|\overrightarrow{\mathrm{a}}\right|=3,\left|\overrightarrow{\mathrm{b}}\right|=4$ and $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=5$, then what is the value of $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|?$

Let $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three mutually perpendicular vectors each of unit magnitude. If $\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}},\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}$, then which one of the following is correct?

If $\overrightarrow{\mathit{a}}$ and $\overrightarrow{b}$ are vectors such that $\left|\overrightarrow{a}\right|=2,$ $\left|\overrightarrow{b}\right|=7$ and $\overrightarrow{a}\times \overrightarrow{b}=3\hat{i}+2\hat{j}+6\hat{k},$ then what is the acute angle between $a$ and $b$?

If $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}=\overrightarrow{0}$ and $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=\lambda (\overrightarrow{b}\times \overrightarrow{c})$, then what is the value of $\lambda$?