Let vector b = λi+4k, λ > 0 and the projection vector of vector b on vector a = 2i+2j-k is vector c.

Question

 Let \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0\) and the projection vector of \(\vec{b}\) on \(\vec{a}=2 \hat{i}+2 \hat{j}-\hat{k}\) is \(\vec{c}.\) If \(|\vec{a}+\vec{c}|=7,\) then the area of the parallelogram formed by vector \(\bar{b}\) and \(\vec{c}\) is (in square units)

(1) 8

(2) 16

(3) 32

(4) 64

Answer 1

Correct option is (3) 32    

\(\bar{c}=(\vec{b} \cdot \hat{a}) \hat{a}=\frac{2 \lambda-4}{6} \vec{a}\)   

\(\because |\vec{a}+\vec{c}|=7 \Rightarrow\left|\vec{a}\left(1+\frac{2 \lambda-4}{9}\right)\right|=7\)  

\(\left|\frac{5+2 \lambda}{9}\right| \times 3=7 \Rightarrow|5+2 \lambda|=21\)  

\(\because \lambda>0 \Rightarrow \lambda=8\)  

\(\Rightarrow \vec{c}=\frac{4}{3} \vec{a}\ \text{ and }\ \vec{b}=4(2 \hat{i}-\hat{k})\)