Question

If $\overrightarrow{a}=\hat{i}+2\hat{j},\overrightarrow{b}=\hat{j}+2\hat{k},$ write a unit vector along the vector $3\overrightarrow{a}-2\overrightarrow{b}.$

Answer 1

Given: $\overrightarrow{a}=\hat{i}+2\hat{j},\overrightarrow{b}=\hat{j}+2\hat{k}$
Therefore,
$$\begin{gathered} 3\overrightarrow{a}-2\overrightarrow{b}=3\hat{i}+6\hat{j}-2\hat{j}-4\hat{k} \\ =3\hat{i}+4\hat{j}-4\hat{k} \end{gathered}$$

Hence, Unit vector along $3\overrightarrow{a}-2\overrightarrow{b}=\frac{3\hat{i}+4\hat{j}-4\hat{k}}{\sqrt{3^2+4^2+{\left(-4\right)}^2}}=\frac{3\hat{i}+4\hat{j}-4\hat{k}}{\sqrt{9+16+16}}=\frac{1}{\sqrt{41}}\left(3\hat{i}+4\hat{j}-4\hat{k}\right)$