First slide

Straight lines in 3D

Question

Equation of the line passing through the point $(\frac{1}{2}, 0, 1)$ and making angles $90 °, 45 °, 45 °$ with axes respectively is

Easy

A

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

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

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

line does not exist

Solution

If $α, β, γ$ are the angles made by a line ‘ $L$ ’ with the coordinate axes in the positive direction then direction cosines are $〈l, m, n〉 = 〈\cos α, \cos β, \cos γ〉$
Hence, the direction cosines of the line which makes angles $90 °, 45 °, 45 °$ with axes respectively is $〈0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}〉$
The direction ratios of the above line are proportional to $〈0, 1, 1〉$
Therefore, the equation of the line passing through the point $(\frac{1}{2}, 0, 1)$ and having the direction ratios $〈0, 1, 1〉$ is $\frac{2 x - 1}{0} = \frac{y - 0}{1} = \frac{z - 1}{1}$

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