Mathematics - Vector Question with Solution | TestHub

$\vec{c}=\left(\frac{\vec{b}\cdot \vec{a}}{|\vec{b}|}\right)\frac{\vec{a}}{|\vec{a}|}$ $\begin{array}{rl}& =\left(\frac{\lambda +8}{9}\right)(\hat{i}+2\hat{j}+2\hat{k})\\ & |\vec{a}+\vec{c}|=7\\ & \Rightarrow \left(\frac{\lambda +8}{9}+1\right)\hat{i}+\left(\frac{2(\lambda +8)}{9}+2\right)\hat{j}+\left(\frac{2(\lambda +8)}{9}+2\right)\hat{k}|=7\\ & {\left(\frac{\lambda +8}{9}+1\right)}^2+{\left(\frac{2(\lambda +8)}{9}+2\right)}^2+{\left(\frac{2(\lambda +8)}{9}+2\right)}^2=49\end{array}$ $\Rightarrow \lambda =4\Rightarrow \vec{c}=\frac{4}{3}\hat{i}+\frac{8}{3}\hat{j}+\frac{8}{3}\hat{k}$ Area of parallelogram $=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{4}{3}& \frac{8}{3}& \frac{8}{3}\\ 4& 0& 4\end{array}\right|=16$