If the lines (x - 3)/1 = (1 - y)/1 = (z + 2)/p and (2 - x)/3 = (y + 1)/5 = (z + 56)/2p are perpendicular to each other, then find the value(s) of p.
For a line in symmetric form (x - x1)/a = (y - y1)/b = (z - z1)/c,
the direction ratios are (a, b, c).
first line
(x - 3)/1 = (1 - y)/1 = (z + 2)/p
rewrite
1 - y as -(y - 1) or (y - 1)/(-1)
So direction ratios first line = (1, -1, p)
for second line (2 - x)/3 = (y + 1)/5 = (z + 56)/2p
Rewrite 2 - x as -(x - 2) so that the first expression changes to (x - 2)/(-3)
So direction ratios of the second line are (-3, 5, 2p)
Two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) are perpendicular if:
a₁×a₂ + b₁×b₂ + c₁×c₂ = 0
(1)(-3) + (-1)(5) + (p)(2p) = 0
-3 - 5 + 2p² = 0
-8 + 2p² = 0
2p² = 8
p² = 4
p = +2 or p = -2
see this video for more explanation
