R S Aggarwal Solutions for Exercise 3: EXERCISE

If $\overrightarrow{\mathit{a}}=(\hat{\mathit{i}}+\hat{\mathit{j}}+\hat{\mathit{k}})\mathit{,}\overrightarrow{\mathit{b}}=(4\hat{\mathit{i}}-2\hat{\mathit{j}}+3\hat{\mathit{k}})$ and $\overrightarrow{c}=(\hat{\mathit{i}}-2\hat{j}+\hat{\mathit{k}})$, find a vector of magnitude $6$ units which is parallel to the vector $(2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c})$.

If the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $\sqrt{3}$ and $2$ respectively, having $\overrightarrow{a}·\overrightarrow{b}=\sqrt{6}$ is $\frac{\mathrm{\pi }}{k}$, then the value of $k$ is

What conclusion can you draw about vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ when $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$ and $\overrightarrow{a}·\overrightarrow{b}=0?$

Write the value of $\hat{i}·(\hat{j}\times \hat{k})+\hat{j}·(\hat{i}\times \hat{k})+\hat{k}·(\hat{i}\times \hat{j})$.

Find the volume of the parallelepiped (in cubic units) whose edges are represented by the vectors $\overrightarrow{a}=(2\hat{i}-3\hat{j}+4\hat{k}),\overrightarrow{b}=(\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{c}=(3\hat{i}-2\hat{j}+2\hat{k})$.

If $\overrightarrow{a}=(2\hat{i}+6\hat{j}+27\hat{k})$ and $\overrightarrow{b}=(\hat{i}+\lambda \hat{j}+\mu \hat{k})$ are such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$, then find the value of $\lambda +\mu$.

If $\frac{\mathrm{\pi }}{k}$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ and $|\overrightarrow{a}·\overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{b}|$, then what is the value of $\frac{k}{2}$?

If $\hat{a}$ and $\hat{b}$ are unit vectors such that $(\hat{a}+\hat{b})$ is a unit vector, the angle between $\hat{a}$ and $\hat{b}$ is $\frac{k\mathrm{\pi }}{3}$, find the value of $k$.