Given: perpendicular drawn from point A (1, 8, 4) to line joining points B (0, -1, 3) and C (2, -3, -1)

To find: foot of perpendicular

Formula Used: Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and with b₁ : b₂ : b₃ being the direction ratios of the line.

If 2 lines of direction ratios a₁:a₂:a₃ and b₁:b₂:b₃ are perpendicular, then a₁b₁+a₂b₂+a₃b₃ = 0

Explanation:

B (0, -1, 3) is a point on the line.

Therefore,

Also direction ratios of the line are (0 - 2) : (-1 + 3) : (3 + 1)

⇒ -2 : 2 : 4

⇒ -1 : 1 : 2

So, equation of the line in Cartesian form is

Any point on the line will be of the form (-λ, λ - 1, 2λ + 3)

So the foot of the perpendicular is of the form (-λ, λ - 1, 2λ + 3)

The direction ratios of the perpendicular is

(-λ - 1) : (λ – 1 - 8) : (2λ + 3 - 4)

⇒ (-λ - 1) : (λ – 9) : (2λ – 1)

From the direction ratio of the line and the direction ratio of its perpendicular, we have

-1(-λ - 1) + λ – 9 + 2(2λ – 1) = 0

⇒ λ + 1 + λ – 9 + 4λ – 2 = 0

⇒ 6λ = 10

So, the foot of the perpendicular is