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

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are vectors such that $\left|\overrightarrow{a}+\overrightarrow{b}\right|=\sqrt{29}$ and $\overrightarrow{a}\times \left(2\hat{i}+3\hat{j}+4\hat{k}\right)=\left(2\hat{i}+3\hat{j}+4\hat{k}\right)\times \overrightarrow{b}$, then a possible value of $\left(\overrightarrow{a}+\overrightarrow{b}\right)·\left(-7\hat{i}+2\hat{j}+3\hat{k}\right)$ is

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\mathrm{\pi }}{3}.$ If $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+\overrightarrow{c},$ where $p, q$ and $r$ are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is:-

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}·\overrightarrow{b}=0.$ For some$x,y\in R$ , let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+(\overrightarrow{a}\times \overrightarrow{b})$. If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$, then the value of $8{\cos }^2\alpha$ is

Consider the cube in the first octant with sides $OP,OQ$ and $OR$ of length $1,$ along the $X-$axis, $Y-$axis and $Z-$axis, respectively, where $O\left(0,0,0\right)$ is the origin. Let $S\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT.$ If $p=SP,q=SQ,r=SR$ and $t=ST$ then the value of $\left|\left(p\times q\right)\times \left(r\times t\right)\right|$ is.........

If $\overrightarrow{a}, \overrightarrow{b}$and $\overrightarrow{c}$ are three non-zero vectors which are pairwise non-collinear. If $\overrightarrow{a}+3\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+2\overrightarrow{c}$ is collinear with $\overrightarrow{a}$ then $\overrightarrow{a}+3\overrightarrow{b}+6\overrightarrow{c}$ is

The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are sides of a triangle $ABC$. The length of the median through $A$ is

Let $\overrightarrow{u}$ be a vector coplanar with the vector $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ & $\overrightarrow{b}=\hat{j}+\hat{k}.$ If $\overrightarrow{u}$ is perpendicular to $\overrightarrow{a}$ and  $\overrightarrow{u}·\overrightarrow{b}=24,$  then${\left|\overrightarrow{u}\right|}^{2 }$is equal to:

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $\left(-{\lambda }^2, 1, 1\right), \left(1, -{\lambda }^2, 1\right)$ and $\left(1, 1, -{\lambda }^2\right)$ also passes through the point $\left(–1, –1, 1\right)$. Then $S$ is equal to: