Let \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0\) and the projection vector of \(\vec{b}\) on \(\vec{a}=2 \hat{i}+…

\(\begin{equation}
\begin{aligned}
& \vec{c}=(\vec{b} \cdot \hat{a}) \hat{a}=\frac{2 \lambda-4}{9} \vec{a} \\
& \because|\vec{a}+\vec{c}|=7 \Rightarrow\left|\vec{a}\left(1+\frac{2 \lambda-4}{9}\right)\right|=7 \\
& \because \lambda>0 \Rightarrow \lambda=8 \\
& \Rightarrow \vec{c}=\frac{4}{3} \vec{a} \text { and } \vec{b}=4(2 \hat{i}+\hat{k}) \\
& \left.\Rightarrow \vec{b} \times \vec{c}=\frac{16}{3}\left|\begin{array}{ll}
\hat{i} & \vec{j} & \vec{k} \\
2 & 0 & 1\\
2 & 2 & -1
\end{array}\right|=\frac{16}{3}(-2 \hat{i}+4 \hat{j}+4 \hat{k}) \right\rvert\, \\
& \Rightarrow|\vec{b} \times \vec{c}|=\frac{32}{3}|-\hat{i}+2 \hat{j}+2 \hat{k}|=32 \\
& \Rightarrow A r e a \text { of parallelogram formed by } \vec{b} \text { and } \vec{c} \\
& \Rightarrow|\vec{b} \times \vec{c}|=32
\end{aligned}
\end{equation}\)