Find the length of the perpendicular drawn from the point (5, 4, −1) to the line $\vec{r} = \hat{i} + λ (2 \hat{i} + 9 \hat{j} + 5 \hat{k}) .$

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Solution

Let the point (5, 4, $-$ 1) be P and the the point through which the line passes be Q (1, 0, 0).
The line is parallel to the vector $\vec{b} = 2 \hat{i} + 9 \hat{j} + 5 \hat{k}$ .

Now,

$\vec{P Q} = - 4 \hat{i} - 4 \hat{j} + \hat{k}$
$∴ \vec{b} \times \vec{P Q} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 9 & 5 \\ - 4 & - 4 & 1 \end{matrix}| = 29 \hat{i} - 22 \hat{j} + 28 \hat{k} \Rightarrow |\vec{b} \times \vec{P Q}| = \sqrt{29^{2} + {(- 22)}^{2} + 28^{2}} = \sqrt{841 + 484 + 784} = \sqrt{2109} |\vec{b}| = \sqrt{2^{2} + 9^{2} + 5^{2}} = \sqrt{4 + 81 + 25} = \sqrt{110} d = \frac{|\vec{b} \times \vec{P Q}|}{|\vec{b}|} = \frac{\sqrt{2109}}{\sqrt{110}} = \sqrt{\frac{2109}{110}}$

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