Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 12: Exercise for Session 8

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves, so that at any time $t$ the position vector $OP$ (where $O$ is the origin) is given by $\hat{a} \cos t+\hat{b} \sin t$, when $P$ is farthest from origin $O$, let $M$ be the length of $OP$ and $\hat{u}$ be the unit vector along $OP$. Then,

Let the vectors $\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.

Statement I: $\overrightarrow{PQ}\times \left(\overrightarrow{RS}+\overrightarrow{ST}\right)\ne 0$, because

Statement II: $\overrightarrow{PQ}\times \overrightarrow{RS}=0$ and $\overrightarrow{PQ}\times \overrightarrow{ST}\ne 0$.

If $\overrightarrow{V}=2\stackrel{⏜}{i}+\stackrel{⏜}{j}-\stackrel{⏜}{k}$ and $\overrightarrow{W}=\stackrel{⏜}{i}+3\stackrel{⏜}{k}$. If $\overrightarrow{U}$ is a unit vector, then the maximum value of the scalar triple product $\left[\overrightarrow{U} \overrightarrow{V} \overrightarrow{W}\right]$ is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}$ and $5\overrightarrow{a}-4\overrightarrow{b}$ are perpendicular to each other, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If the vectors $p\hat{i}+\hat{j}+\hat{k}, \hat{i}+q\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r\hat{k}$ (where $p\ne q\ne r\ne 1)$) are coplanar, then the value of $pqr-(p+q+r)$ is

If the vectors $\overrightarrow{a}=\hat{i}-\hat{j}+2\hat{k}, \overrightarrow{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+u\hat{k}$ are mutually orthogonal, then $\left(\lambda ,\mu \right)$ is

A tetrahedron has vertices at $O\left(0,0,0\right), A\left(1,2,1\right), B\left(2,1,3\right)$ and $C\left(–1,1,2\right)$. Then, the angle between the faces $OAB$ and $ABC$ will be

Given, two vectors are $\hat{i}-\hat{j}$ and $\hat{i}+2\hat{j},$ the unit vector coplanar with the two vectors and perpendicular to first is