Embibe Experts Solutions for Chapter: Mathematical Tools in Physics, Exercise 1: Exercise-1

If $\left|\overrightarrow{A}\right|=4, \left|\overrightarrow{B}\right|=3$ and $\theta =60°$ in the figure, Find $\left(a\right) \overrightarrow{A}·\overrightarrow{B}, \left(b\right) \left|\overrightarrow{A}\times \overrightarrow{B}\right|$

If $\overrightarrow{A}=4\hat{\mathrm{i}}+n\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ and $\overrightarrow{B}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}},$ then find the value of $n$ so that $\overrightarrow{A}\perp \overrightarrow{B}.$

Find a unit vector perpendicular to both the vectors $\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$ and $\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right).$

Area of a parallelogram, whose diagonals are $3\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ will be:

If $\overrightarrow{A}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\overrightarrow{B}=\hat{\mathrm{i}}-\hat{\mathrm{j}}$, the value of $\left(\overrightarrow{A}+\overrightarrow{B}\right)·\left(\overrightarrow{A}-\overrightarrow{B}\right)$ is:

Vectors $\overrightarrow{A}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{B}=3\hat{i}+3\hat{j}-6\hat{k}$ are :

If two vectors are given as :
$\overrightarrow{A}=\hat{i}+2\hat{j}-\hat{k}$ and $\overrightarrow{B}=-\hat{i}-\hat{j}+\hat{k}$ then the vector which is not perpendicular to $\left(\overrightarrow{A}\times \overrightarrow{B}\right)$ is:

The vector sum of two forces is perpendicular to their vector differences. In that case, the forces: