Amit M Agarwal Solutions for Chapter: Vector Algebra, Exercise 6: Target Exercises

If the non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other, then the solution of the equation, $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}$ is given by

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

If $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$ and $\overrightarrow{r}·\overrightarrow{a}=0,$ where $\overrightarrow{a}=2\hat{i}+3\hat{j}-\widehat{\mathit{k}}$, $\overrightarrow{b}=3\hat{\mathit{i}}-\hat{\mathit{j}}+\widehat{\mathit{k}}$ and $\overrightarrow{c}=\hat{\mathit{i}}+\hat{\mathit{j}}+3\widehat{\mathit{k}},$ then $\overrightarrow{r}$ is equal to

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are any three non-zero and non-coplanar vectors and $x=\frac{\left|\overrightarrow{r} \overrightarrow{b} \overrightarrow{c}\right|}{\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]}, y=\frac{\left|\overrightarrow{r} \overrightarrow{\mathrm{c}} \overrightarrow{a}\right|}{\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]} \& z=\frac{\left|\overrightarrow{r} \overrightarrow{a} \overrightarrow{b}\right|}{\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]}$, then any vector $\overrightarrow{r}$ is equal to

If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{r}$ is a non-zero vector such that $p\overrightarrow{r}+\left(\overrightarrow{r}·\overrightarrow{b}\right)\overrightarrow{a}=\overrightarrow{c},$ then $\overrightarrow{r}$ is equal to

If $\overrightarrow{p}=\lambda \left(\overrightarrow{u}\times \overrightarrow{v}\right)+\mu \left(\overrightarrow{v}\times \overrightarrow{w}\right)+\nu \left(\overrightarrow{w}\times \overrightarrow{u}\right)$ and $\left[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}\right]=\frac{1}{5},$ then $\lambda +\mu +\nu$ is equal to

If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\widehat{k}, \overrightarrow{b}=-\hat{i}+2\hat{j}+\widehat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}$, then $\overrightarrow{a}+t\overrightarrow{b}$ is perpendicular to $\overrightarrow{c},$ if $t$ is equal to

If the vectors $\overrightarrow{b}=\left(\tan \alpha ,-1,2\sqrt{\sin \frac{\alpha }{2}}\right)$ and $\overrightarrow{c}=\left(\tan \alpha ,\tan \alpha ,-\frac{3}{\sqrt{\sin \frac{\alpha }{2}}}\right)$ are orthogonal and a vector $\overrightarrow{a}=\left(1,3,\sin 2\alpha \right)$ makes an obtuse angle with the $Z$-axis, then the value of $\alpha$ is