Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 3: Level 3

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=p\overrightarrow{d}, \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}=q\overrightarrow{a}$ and $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar, then $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}$ is equal to

Let $\overrightarrow{u}$ be a vector on rectangular coordinate system with sloping angle $60°$. Suppose that, $\left|\overrightarrow{u}-\hat{i}\right|$ is geometric mean of $\left|\overrightarrow{u}\right|$ and $\left|\overrightarrow{u}-2\hat{i}\right|$ where $\hat{i}$ is the unit vector along $x$-axis then $\left|\overrightarrow{u}\right|$ has the value equal to $\sqrt{a}-\sqrt{b}$ where $a, b\in N.$ Then the value of ${\left(a+b\right)}^3+{\left(a-b\right)}^3$ is

If $\left|\overrightarrow{a}+\overrightarrow{b}\right|<\left|\overrightarrow{a}-\overrightarrow{b}\right|,$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ can lie in the interval:

$P\left(\overrightarrow{\mathit{p}}\right)$ and $Q\left(\overrightarrow{\mathit{q}}\right)$ are the position vectors of two fixed points and $R\left(\overrightarrow{r}\right)$ is the position vector of a variable point. If $R$ moves such that $\left(\overrightarrow{r}-\overrightarrow{p}\right)\times \left(\overrightarrow{r}-\overrightarrow{q}\right)=\overrightarrow{0},$ then the locus of $R$ is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are any two vectors of magnitudes $1$ and $2,$ respectively, and ${\left(1-3\overrightarrow{a}·\overrightarrow{b}\right)}^2+{\left|2\overrightarrow{a}+\overrightarrow{b}+3\left(\overrightarrow{a}\times \overrightarrow{b}\right)\right|}^2=47,$ then the angle between$\overrightarrow{a}$ and $\overrightarrow{\mathrm{b}}$ is

If $\overrightarrow{a}=\alpha \hat{i}+\beta \hat{j}+3\hat{k}, \overrightarrow{b}=-\beta \hat{i}-\alpha \hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{i}-2\hat{j}-\hat{k}$ such that $\overrightarrow{a}·\overrightarrow{b}=1$ and $\overrightarrow{b}·\overrightarrow{c}=-3,$ then $\frac{1}{3}\left(\right(\overrightarrow{a}\times \overrightarrow{b})·\overrightarrow{c})$ is equal to _______.

Given that $\overrightarrow{a}, \overrightarrow{b} , \overrightarrow{p}, \overrightarrow{q}$ are four vectors such that $\overrightarrow{a}+\overrightarrow{b}=\mu \overrightarrow{p},\overrightarrow{ b}·\overrightarrow{q}=0$ and ${\left|\overrightarrow{b}\right|}^2=1,$ where $\mu$ is a scalar. Then $\left|\left(\overrightarrow{a}·\overrightarrow{q}\right)\overrightarrow{p}-\left(\overrightarrow{p}·\overrightarrow{q}\right)\overrightarrow{a}\right|$is equal to:

Let $\hat{a}$ and $\hat{b}$ be mutually perpendicular unit vectors. Then for any arbitrary $\hat{r},$