Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 1: Exercise for Session 1

If  $\overrightarrow{a}=\hat{\mathit{i}}+2\hat{j}-2\hat{k}, \overrightarrow{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-\hat{k},$ then $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is equal to

The horizontal force and the force inclined at an angle ${60}^{\circ }$with the vertical, whose resultant is in vertical direction of $P \mathrm{kg}$, are

Given a parallelogram $ABCD$. If $\left|\overrightarrow{AB}\right|=a, \left|\overrightarrow{AD}\right|=b, \left|\overrightarrow{AC}\right|=c,$ then $\overrightarrow{DB}·\overrightarrow{AB}$ has the value

For two particular vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, it is known that  $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$. What must be true about the two vectors?

For some non- zero vector $\overrightarrow{V}$, if the sum of $\overrightarrow{V}$ and the vector obtained from $\overrightarrow{V}$ by rotating it by $\angle 2\alpha$ equals to the vector obtained from $\overrightarrow{V}$ by rotating it by  $\angle \alpha$, then the value of $\alpha$ is

In the isosceles $△ABC$, $\left|\overrightarrow{\mathrm{AB}}\right|=\left|\overrightarrow{\mathrm{BC}}\right|=8$, a point $E$ divides $AB$ internally in the ratio $1:3$, then the cosine of the angle between $CE$ and $CA$ is (where, $\left|\overrightarrow{\mathrm{CA}}\right|=12$)

Given an equilateral $∆ABC$ with side length equal to $a$. Let $M$ & $N$ be two points respectively on the side $\overrightarrow{AB}$ and $\overrightarrow{AC}$ such that $\overrightarrow{AN}=k\overrightarrow{AC}$ and $\overrightarrow{AM}=\frac{\overrightarrow{AB}}{3}$. If  $\overrightarrow{BN}$ and $\overrightarrow{CM}$ are orthogonal, then the value of $k$ is

The vector $\overrightarrow{c}$ is perpendicular to the vectors $\overrightarrow{a}=\left(2,-3,1\right)$, $\overrightarrow{b}=\left(1,-2,3\right)$ and satisfies the condition $\overrightarrow{c}·\left( \hat{\mathrm{i}}+2\hat{\mathrm{j}}-7\hat{\mathrm{k}}\right)=10$.Then, the vector $\overrightarrow{c}$ is equal to