Question

Find the perpendicular distance of the point (1, 0, 0) from the line `(x - 1)/(2) = (y + 1)/(-3) = (z + 10)/(8)` Also find the co-ordinates of the foot of the perpendicular.

Answer 1

Let foot of the perpendicular 'M' is drawn from point p(1, 0, 0) the line L: `(x - 1)/(2) = (y + 1)/(-3) = (z + 10)/(8)`

∴ x₁ = 1, y₁ = −1, z₁ = −10

Direction ratios are 2, 3, -8

let L1: `(x - 1)/(2) = (y + 1)/(-3) = (z + 10)/(8)`

∴ Parametrics co-ordinates (2λ + 1, −3λ − 1, 8λ −10)

∴ The direction ratio of PM is (2λ, −3λ − 1, 8λ −10)

∴ PM is ⊥ to the given line` (2λhati(−3λ −1)hatj + (8λ −10)hatk) * (2hati - 3hatj + 8hatk) = 0`

∴ 4λ + 9λ + 3 + 64λ − 80 = 0

∴ 77λ − 77 = 0

∴ λ = 1

∴ Co-ordinates of the foot of the perpendicular are (3, −4, −2)

∴ d = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2`

= `sqrt((2)^2 - (-4)^2+ (-2)^2`

= `sqrt (4 + 16 + 4)`

= `sqrt24`

∴ d = `2sqrt6`