Let c be the projection vector of b = i +4 k , >0, on the vector … - Vidyadip

Question

Let $\overrightarrow{ c }$ be the projection vector of $\overrightarrow{ b }=\lambda \hat{ i }+4 \hat{ k }, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\overrightarrow{ c }$ is ___________ .

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Answer

16
$
\begin{array}{l}Sol.
\overrightarrow{c}=\left(\frac{\overrightarrow{b} \cdot \overrightarrow{a}}{|\overrightarrow{a}|}\right) \frac{\overrightarrow{a}}{|\overrightarrow{a}|} \\
=\left(\frac{\lambda+8}{9}\right)(\hat{i}+2 \hat{j}+2 \hat{k}) \\
|\overrightarrow{a}+\overrightarrow{c}|-=7 \Rightarrow \lambda=4
\end{array}
$
Area of parallelogram
$\begin{array}{l}=|\overrightarrow{ b } \times \overrightarrow{ c }|=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ \frac{4}{3} & \frac{8}{3} & \frac{8}{3} \\ 4 & 0 & 4\end{array}\right| \\ =16\end{array}$

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