If a = i - j - k and b = i - 3 j + k and the orthogonal projectio… - Vidyadip

MCQ

If $\vec a = \hat i - \hat j - \hat k$ and $\vec b = \lambda \hat i - 3\hat j + \hat k$ and the orthogonal projection of $\vec b$ on $\vec a$ is $\frac{4}{3}\left( {\hat i - \hat j - \hat k} \right)$, then $\lambda$ is equal to

A
$0$
✓
$2$
C
$12$
D
$-1$

✓

Answer

Correct option: B.

$2$

b
$\frac{{(\overrightarrow b \cdot \overrightarrow a )\overrightarrow a }}{{|\overrightarrow a {|^2}}} = \frac{4}{3}(\widehat {\rm{i}} - \widehat {\rm{j}} - \widehat {\rm{k}})$

$ \Rightarrow \frac{{\{ (\lambda \widehat i - 3\widehat j + \widehat k) \cdot (\widehat i - \widehat j - \widehat k)\} (\widehat i - \widehat j - \widehat k)}}{{(1 + 1 + 1)}}$

$=\frac{4}{3}(\hat{i}-\hat{j}-\hat{k})$

$\Rightarrow(\lambda+3-1)(\hat{i}-\hat{j}-\hat{k})=4(\hat{i}-\hat{j}-\hat{k})$

$\Rightarrow(\lambda+2)(\hat{i}-\hat{j}-\hat{k})=4(\hat{i}-\hat{j}-\hat{k})$

On equating the coefficient of $\widehat i$, we get

$\lambda+2=4 \Rightarrow \lambda=2$

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