First slide

Dot product or scalar product of vectors

Question

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

Moderate

A

$2 \hat{i} + 3 \hat{j} - 3 \hat{k}$
B

$2 \hat{i} + 3 \hat{j} + 3 \hat{k}$
C

$2 \hat{i} + \hat{j} + 5 \hat{k}$
D

$5 i - 6 j - 7 k$

Solution

$Let \vec{d} = \vec{b} + λ \vec{c}$
$Then length of the projection of \bar{d} on \bar{a} is \frac{\bar{a} \cdot \bar{d}}{\sqrt{a}}$
$⇒ \frac{\vec{a} \cdot \vec{d}}{| \vec{a} |} = \pm \sqrt{\frac{2}{3}} \Rightarrow λ = 1$
$\Rightarrow \vec{d} = 2 \hat{i} + 3 \hat{j} - 3 \hat{k}$

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