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

If the vectors $a\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+c\hat{\mathrm{k}}$ $\left(a,b,c\ne 1\right)$ are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

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 $\overrightarrow{\alpha }=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\beta }=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ 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 $\overrightarrow{\delta }$ ?

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 $b$?

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?

Three vectors $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are such that $\left|\overrightarrow{a}\right|=1, \left|\overrightarrow{b}\right|=2, \left|\overrightarrow{c}\right|=4$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$. Then, the value of $4\overrightarrow{a}·\overrightarrow{b}+3\overrightarrow{b}·\overrightarrow{c}+3\overrightarrow{c}·\overrightarrow{a}$ is equal to