When are the two lines x = ay + b, z = cy + d | vedclass

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Question

(A) The first line is given by $x = ay + b$ and $z = cy + d$. This can be written in symmetric form as $\frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c

Vedclass · Accepted Answer

$aa' + cc' + 1 = 0$

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(A) The first line is given by $x = ay + b$ and $z = cy + d$. This can be written in symmetric form as $\frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c}$. The direction ratios of this line are $(a, 1, c)$.The second line is given by $x = a'y + b'$ and $z = c'y + d'$. This can be written in symmetric form as $\frac{x - b'}{a'} = \frac{y}{1} = \frac{z - d'}{c'}$. The direction ratios of this line are $(a', 1, c')$.Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.Applying this condition to our lines: $(a)(a') + (1)(1) + (c)(c') = 0$.Therefore,$aa' + cc' + 1 = 0$.

When are the two lines $x = ay + b, z = cy + d$ and $x = a'y + b', z = c'y + d'$ perpendicular to each other?

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