Q.170. Lines in R^3.

Question 170.
Find the angle between the lines whose direction cosines are given by the equations,
l + m + n = 0 and l^2 + m^2 - n^2 = 0.

Answer 170.
l + m + n = 0
=> n = - (l + m).
Plugging in the second equation,
l^2 + m^2 - (l + m)^2 = 0
=> 2lm = 0
=> l = 0 or m = 0

If l = 0, m + n = 0 => m = - n
=> (0, - n, n ) are the direction cosines of one of the two lines

If m = 0, l + n = 0 => l = - n
=> (- n, 0, n) are the direction cosines of the second line.

If θ = angle between the two lines,
cosθ = [(0, - n, n) . (- n, 0, n)] / [l (0, - n, n) l * l (- n, 0, n) l]
=> cosθ = (0*-n - n*0 + n*n) / [√(n^2 + n^2) * √(n^2 + n^2)]
=> cosθ = n^2 / 2n^2 = 1/2
=> θ = π/3.

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