(a)OR(b)

Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\displaystyle \vec{a}=3 \hat{i}+\hat{j}+2 \hat{k}$ and $\displaystyle \vec{b}=2 \hat{i}-2 \hat{j}+4 \hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.

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