First slide

Straight lines in 3D

Question

The condition that the straight lines $x = a_{1} y + b_{1}; z = c_{1} y + d_{1}$ and $x = a_{2} y + b_{2}; z = c_{2} y + d_{2}$ are perpendicular, is

Easy

A

$a_{1} a_{2} + c_{1} c_{2} = 1$
B

$a_{1} a_{2} + c_{1} c_{2} = - 1$
C

$a_{1} a_{2} + c_{1} c_{2} = 2$
D

$a_{1} a_{2} + c_{1} c_{2} = 0$

Solution

The given lines rewrite in symmetric form as $\frac{x - b_{1}}{a_{1}} = y = \frac{z - d_{1}}{c_{1}}$ and $\frac{x - b_{2}}{a_{2}} = y = \frac{z - d_{2}}{c_{2}}$
The above two lines are perpendicular,
hence $a_{1} a_{2} + 1 + c_{1} c_{2} = 0$

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