Question

Let $a=2i-j+k,b=i+2j-k$ and $c=i+j-2k$ be three vectors. A vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

• A. $2i+3j-3k$
• B. $2i+3j+3k$
• C. $2i-j+5k$
• D. $2i+j+5k$

Answer 1

Answer: (A), (C)

$2i-j+5k$
$2i+3j-3k$

Any vector in the plane $b$ and $c$ is

$r=b+\lambda c=(i+2j+k)+\lambda (i+j-2k)=(1+\lambda )i+(2+\lambda )j+(-1-2\lambda )k$

Projection of $r$ on $a$ is $=\frac{r.a}{\left|a\right|}$

$=\frac{2(1+\lambda )-(2+\lambda )-(1+2\lambda )}{\sqrt{4+1+1}}=\frac{-\lambda -1}{\sqrt{6}}$

$\therefore \frac{-\lambda -1}{\sqrt{6}}=\pm \sqrt{\frac{2}{3}}\Rightarrow -\lambda -1=\pm 2\Rightarrow \lambda =-3,1$

Hence $r=-2i-j+5k$ and $r=2i+3j-3k$