The vector a = - i ^ + 2 j ^ + k ^ is rotated through a right angle, passing through the y -axis in its way and the resulting vector is b . Then the projection of 3 a + 2 ? b on c = 5 i ^ + 4 j ^ + 3 k ^ is [2023]

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Solution

Q.
The vector $\vec{a} = - \hat{i} + 2 \hat{j} + \hat{k}$ is rotated through a right angle, passing through the $y$ -axis in its way and the resulting vector is $\vec{b}$ . Then the projection of $3 \vec{a} + \sqrt{2} \vec{b}$ on $\vec{c} = 5 \hat{i} + 4 \hat{j} + 3 \hat{k}$ is [2023]

1 $3 \sqrt{2}$

2 $1$

3 $2 \sqrt{3}$

4 $\sqrt{6}$

Ans.
(1)

$\vec{b}, \vec{a} and \hat{j} are coplanar.$

$\vec{b} = λ \vec{a} + μ \hat{j} = λ (- \hat{i} + 2 \hat{j} + \hat{k}) + μ \hat{j}$

$= - λ \hat{i} + (2 λ + μ) \hat{j} + λ \hat{k},$ $\vec{b} \cdot \vec{a} = 0$

$(- \hat{i} + 2 \hat{j} + \hat{k}) \cdot (- λ \hat{i} + (2 λ + μ) \hat{j} + λ \hat{k}) = 0$

$λ + 2 (2 λ + μ) + λ = 0 \Rightarrow 6 λ + 2 μ = 0 \Rightarrow μ + 3 λ = 0$

$\vec{b} = λ \vec{a} - 3 λ \hat{j} = λ (- \hat{i} + 2 \hat{j} + \hat{k}) - 3 λ \hat{j}$ $= λ (- \hat{i} - \hat{j} + \hat{k})$

$| \vec{b} | = \sqrt{3} | λ | = \sqrt{6} [∵ | \vec{a} | = \sqrt{6}] \Rightarrow | λ | = \sqrt{2} \Rightarrow λ \neq \sqrt{2}$

As for this value of $λ$ angle between b and y-axis is not acute.

$Therefore λ = - \sqrt{2};$

$3 \vec{a} + \sqrt{2} \vec{b} = - \hat{i} + 8 \hat{j} + \hat{k};$ $| \vec{c} | = 5 \sqrt{2}$

$\frac{(3 \vec{a} + \sqrt{2} \vec{b}) \cdot \vec{c}}{| \vec{c} |} = \frac{(- \hat{i} + 8 \hat{j} + \hat{k}) \cdot (5 \hat{i} + 4 \hat{j} + 3 \hat{k})}{5 \sqrt{2}}$

$= \frac{- 5 + 32 + 3}{5 \sqrt{2}} = \frac{30}{5 \sqrt{2}} = \frac{6}{\sqrt{2}} = 3 \sqrt{2}$

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