# Find the coordina

Given: perpendicular drawn from point A (1, 2, 1) to line joining points B (1, 4, 6) and C (5, 4, 4)

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 (1, 4, 6) is a point on the line.

Therefore, Also direction ratios of the line are (1 - 5) : (4 – 4) : (6 – 4)

-4 : 0 : 2

-2 : 0 : 1

So, equation of the line in Cartesian form is Any point on the line will be of the form (-2λ + 1, 4, λ + 6)

So the foot of the perpendicular is of the form (-2λ + 1, 4, λ + 6)

The direction ratios of the perpendicular is

(-2λ + 1 – 1) : (4 - 2) : (λ + 6 - 1)

(-2λ) : 2 : (λ + 5)

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

-2(-2λ) + 0 + λ + 5 = 0

4λ + λ = -5

λ = -1

So, the foot of the perpendicular is (3, 4, 5)

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