A (1, 0, 4), B(0, – 11, 3), C(2, – 3, 1) are three points, and D is the foot of the perpendicular from A on BC. Find the coordinates of D.

Given: - Perpendicular from A(1, 0, 4) drawn at line joining points B(0, – 11, 3) and C(2, – 3, 1)


and D be the foot of the perpendicular drawn from A(1, 0, 4) to line joining points B(0, – 11, 3) and C(2, – 3, 1).


Now let's find the equation of the line which is formed by joining points B(0, – 11, 3) and C(2, – 3, 1)


Tip: - Equation of a line joined by two points A(x1,y1,z1) and B(x2,y2,z2) is given by





Now



Therefore,


⇒ x = 2λ, y = 8λ – 11, z = – 2λ + 3


Therefore, coordinates of D(2λ, 8λ – 11, – 2λ + 3)


Now as we know (TIP) ‘if two points A(x1,y1,z1) and B(x2,y2,z2) on a line, then its direction ratios are proportional to (x2 – x1,y2 – y1,z2 – z1)’


Hence


Direction Ratios of AD


= (2λ – 1), (8λ – 11 – 0), ( – 2λ + 3 – 4)


= (2λ – 1), (8λ – 11), ( – 2λ – 1)


and by comparing with given line equation, direction ratios of the given line are


(hint: denominator terms of line equation)


= (2,8, – 2)


Since the AD is perpendicular to given line, therefore by “condition of perpendicularity.”


a1a2 + b1b2 + c1c2 = 0 ; where a terms and b terms are direction ratio of lines which are perpendicular to each other.


⇒ 2(2λ – 1) + (8)(8λ – 11) – 2( – 2λ – 1) = 0


⇒ 4λ – 2 + 64λ – 88 + 4λ + 2 = 0


⇒ 72λ – 88 = 0




Therefore coordinates of D


i.e. Foot of perpendicular


By putting the value of λ in D coordinate equation, we get





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