Question

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l² = m² + n² is ______.

विकल्प

• `pi/6`

• `pi/2`

• `pi/3`

• `pi/4`

Answer 1

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l² = m² + n² is `pi/3`.

Explanation:

Putting l = – m – n in l² = m² + n², we get

(–m – n)² = m² + n²

⇒ mn = 0

⇒ m = 0 or n = 0

If m = 0, then l = – n

∴ `l/(-1) = "m"/0 = "n"/1`

If n = 0, then l = – m

∴ `l/(-1) = "m"/1 = "n"/0`

∴ a₂, b₁, c₁ = –1, 0, 1 and a₁, b₂, c₂ = –1, 1, 0

∴ The angle between the lines is given by

cos θ = `(1 + 0 + 0)/(sqrt(1 + 0 + 1)sqrt(1 + 1 + 0)) = 1/2`

∴ θ = `pi/3`