First slide

Straight lines in 3D

Question

The direction cosines of the line $\frac{x - 1}{l} = \frac{y + 1}{l + 1} = \frac{z - 1}{l}$

Very Easy

A

$⟨ \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} ⟩$
B

$⟨ 0, - 1, - 0 ⟩$
C

$⟨ 1, 0, - 1 ⟩$
D

$⟨ - \frac{2}{3}, \frac{1}{3}, - \frac{2}{3} ⟩$

Solution

If $⟨ l, m, n ⟩$ are direction cosines of a line then $l^{2} + m^{2} + n^{2} = 1$
Hence, $l^{2} + {(l + 1)}^{2} + l^{2} = 1$
It implies that $3 l^{2} + 2 l = 0 \Rightarrow l (3 l + 2) = 0 \Rightarrow l = 0, - \frac{2}{3}$
Therefore, the direction cosines are $⟨ - \frac{2}{3}, \frac{1}{3}, - \frac{2}{3} ⟩$

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