Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 11: Exercise for Session 7

A non-zero vector $\overrightarrow{\mathrm{a}}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{\mathrm{i}},\hat{\mathrm{i}}+\hat{\mathrm{j}}$  and $\hat{\mathrm{i}}-\hat{\mathrm{j}}, \hat{\mathrm{j}}+\hat{\mathrm{k}}$ Find the angle between $\overrightarrow{\mathrm{a}}$ and $\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$.

The vectors  $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ turns through a right angle while passing through the positive $x$–axis on the way. Find the vector in its new position.

Let $\hat{\mathrm{u}}$ and $\hat{\mathrm{v}}$  are unit vectors and $\overrightarrow{\mathrm{w}}$ is a vector such that $\hat{\mathrm{u}}\text{×}\hat{\mathrm{v}}+\hat{\mathrm{u}}=\overrightarrow{\mathrm{w}}$ and $\overrightarrow{\mathrm{w}}\times \hat{\mathrm{u}}=\hat{\mathrm{v}}$, then find the value of $[\hat{\mathrm{u}}\hat{\mathrm{v}}\hspace{0.33em}\overrightarrow{\mathrm{w}}].$

If $\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{C}}$ are three vectors given by $2\hat{\mathrm{i}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $4\hat{\mathrm{i}}-3\hat{\mathrm{j}}+7\hat{\mathrm{k}}$.  Then, find $\overrightarrow{\mathrm{R}}$ which satisfies the relation $\overrightarrow{\mathrm{R}}\times \overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{C}}\times \overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{R}}.\overrightarrow{\mathrm{A}}=0.$

If $\overrightarrow{x}·\overrightarrow{a}=\overrightarrow{x}·\overrightarrow{b}=1, [\overrightarrow{x} \overrightarrow{a} \overrightarrow{b}]=1$ and $\overrightarrow{a}·\overrightarrow{b}=0,$ then find $\overrightarrow{x}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$.

Let $\hat{x}, \hat{y}$ and $\hat{z}$ be unit vectors such that $\hat{\mathrm{x}}+\hat{\mathrm{y}}+\hat{\mathrm{z}}=\overrightarrow{\mathrm{a}}, \hat{\mathrm{x}}\times (\hat{\mathrm{y}}\times \hat{\mathrm{z}})=\overrightarrow{b}, (\hat{\mathrm{x}}\times \hat{\mathrm{y}})\times \hat{\mathrm{z}}=\overrightarrow{c}, \overrightarrow{\mathrm{a}}·\hat{\mathrm{x}}=\frac{3}{2}, \overrightarrow{\mathrm{a}}.\hat{\mathrm{y}}=\frac{7}{4}$ and $\left|\overrightarrow{\mathrm{a}}\right|=2$. Find $\hat{x}, \hat{y}$ and $\hat{z}$  in terms of $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$.

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three mutually perpendicular vectors of equal magnitude. If the vector $\overrightarrow{x}$satisfies the equation $\overrightarrow{a}\times \left\{(\overrightarrow{x}-\overrightarrow{b})\times \overrightarrow{a}\right\}+ \overrightarrow{b}\times \left\{(\overrightarrow{x}-\overrightarrow{c})\times \overrightarrow{b}\right\}+\overrightarrow{c}\times \left\{(\overrightarrow{x}-\overrightarrow{a})\times \overrightarrow{c}\right\}=0$, then find $\overrightarrow{x}$.

Given vectors $\overrightarrow{CB}=\overrightarrow{a}, \overrightarrow{CA}=\overrightarrow{b}$ and $\overrightarrow{CO}=\overrightarrow{x}$, where $O$ is the centre of the circle circumscribed $∆ABC$, then find vector $\overrightarrow{x}$.