(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by…

Let the required angle between the kite strings be $\displaystyle \boldsymbol{\theta}$. \[\begin{aligned} & \text { Then, } \cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \\ & \Rightarrow \cos \theta==\frac{(3 \hat{i}+\hat{j}+2 \hat{k})(2 \hat{i}-2 \hat{j}+4 \hat{k})}{\sqrt{9+1+4} \sqrt{4+4+16}}=\frac{12}{\sqrt{336}}=\frac{3}{\sqrt{21}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{12}{\sqrt{336}}\right) \operatorname{orcos}^{-1}\left(\frac{3}{\sqrt{21}}\right) \end{aligned} \] \[\begin{aligned} & \overrightarrow{\mathrm{B} \mathrm{A}}=-6 \hat{i}+2 \hat{j}+3 \hat{k} \\ & \text { Requiredunit vector of magnitude } 21 \\ & =21 \times\left(\frac{-6 \hat{i}+2 \hat{j}+3 \hat{k}}{\sqrt{36+4+9}}\right) \\ & =3(-6 \hat{i}+2 \hat{j}+3 \hat{k}) \text { or }-18 \hat{i}+6 \hat{j}+9 \hat{k} \end{aligned} \]