First slide

Straight lines in 3D

Question

Equation of line passing through $(1, - 1, 0)$ and parallel to the line having direction cosines $〈\frac{\sqrt{3}}{2}, 0, \frac{1}{2}〉$ is

Easy

A

$\frac{x - \frac{\sqrt{3}}{2}}{1} = \frac{y - 0}{- 1} = \frac{z - \frac{1}{2}}{0}$
B

$\frac{x - 1}{\sqrt{3}} = \frac{y + 1}{0} = \frac{z}{1}$
C

$\frac{2 x}{\sqrt{3}} = \frac{y}{0} = \frac{2 z}{1}$
D

$\frac{x + y}{- 3} = \frac{y - 1}{2} = \frac{z - 3}{1}$

Solution

The equation of the line passing through the point $(x_{1}, y_{1}, z_{1})$ and parallel to the line whose direction ratios are $〈a, b, c〉$ is $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$
Here the direction cosines are $〈\frac{\sqrt{3}}{2}, 0, \frac{1}{2}〉$ , it implies that the direction ratios are $〈\sqrt{3}, 0, 1〉$
Therefore, the equation of the line passing through the point $(1, - 1, 0)$ and having direction ratios $〈\sqrt{3}, 0, 1〉$ is $\frac{x - 1}{\sqrt{3}} = \frac{y + 1}{0} = \frac{z}{1}$

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