First slide

Straight lines in 3D

Question

$The distance of the point - \bar{i} + 2 \bar{j} + 6 \bar{k} from the straight line through the point (2, 3, - 4) and$ $parallel to 6 \bar{i} + 3 \bar{j} - 4 \bar{k} is$

Moderate

A

5
B

6
C

7
D

8

Solution

[SolutionStep1]:
From the figure
$Dr's of \vec{A P} = (- 3, - 1, 10) (x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$
[SolutionStep2]:
$|\vec{A P}| = \sqrt{110}$
$\bar{P N} vector is perpendicular to the line$
[SolutionStep3]:
$\vec{A N} = The projection of \vec{A P} one the line (on (6, 3, - 4))$
$= |\frac{\vec{A P} . (6, 3, - 4)}{(6, 3, - 4)}|$
$= |\frac{(- 3, - 1, 10) . (6, 3, - 4)}{(6, 3, - 4)}|$
$= \frac{|- 18 - 3 - 40|}{\sqrt{61}} = \sqrt{61}$
$From figure P N^{2} = A P^{2} - A N^{2}$
$= 110 - 61 = 49$
$\Rightarrow P N = 7$

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