Andhra Pradesh Board Solutions for Exercise 2: Exercise 5 (b)

If $\overline{a}=2\hat{i}+3\hat{j}+4\hat{k}, \overline{b}=\hat{i}+\hat{j}-\hat{k} \text{and }\overline{c}=\hat{i}-\hat{j}+\hat{k}$, then compute $\left(\overline{a}\times \left(\overline{b}\times \overline{c}\right)\right)$ and verify that it is perpendicular to $\overline{a}$.

If $\overrightarrow{a}=7\hat{i}-2\hat{j}+3\hat{k}, \overrightarrow{b}=2\hat{i}+8\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}+\hat{k}$, then compute $\overrightarrow{a}\times \overrightarrow{b}, \overrightarrow{a}\times \overrightarrow{c}$ and $\overrightarrow{a}\times \left(\overrightarrow{b}+\overrightarrow{c}\right)$, verify whether the cross product is distributive over vector addition.

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{c}=\hat{j}-\hat{k}$, then find $\overrightarrow{b}$ such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$ and $\overrightarrow{a}·\overrightarrow{b}=3$.

$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three vectors of equal magnitudes and each of them is inclined at an angle of $60°$ to the others. If $\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|=\sqrt{6}$, then find $\left|\overrightarrow{a}\right|$. 

For any two vectors $\overrightarrow{a} \mathrm{and} \overrightarrow{b}$, show that $\left(1+|\overrightarrow{a}{|}^2\right)\left(1+{\left|\overrightarrow{b}\right|}^2\right)=|1-\overrightarrow{a}·\overrightarrow{b}{|}^2+|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}{|}^2$.

If $\overrightarrow{a}, \overrightarrow{b},\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}$ is perpendicular to the plane of $\overrightarrow{b}, \overrightarrow{c}$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}$, then find $\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|$.

If $\overrightarrow{a}=3\hat{i}-\hat{j}+2\hat{k}, \overrightarrow{b}=-\hat{i}+3\hat{j}+2\hat{k} , \overrightarrow{c}=4\hat{i}+5\hat{j}-2\hat{k} \mathrm{and} \overrightarrow{d}=\hat{i}+3\hat{j}+5\hat{k}$, then compute $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})$.

If $\overrightarrow{a}=3\hat{i}-\hat{j}+2\hat{k}, \overrightarrow{b}=-\hat{i}+3\hat{j}+2\hat{k} , \overrightarrow{c}=4\hat{i}+5\hat{j}-2\hat{k} \mathrm{and} \overrightarrow{d}=\hat{i}+3\hat{j}+5\hat{k}$, then compute$(\overrightarrow{a}\times \overrightarrow{b})·\overrightarrow{c}-(\overrightarrow{a}\times \overrightarrow{d})\cdot \overrightarrow{b}$ .