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Question
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
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Solution
Let L be the foot of perpendicular drawn from the points A (1, 8, 4) to the line passing through B and C as shown in the Figure.
The equation of line BC by using formula `vec"r" = vec"a" + lambda(vec"b" - vec"a")`, the equation of the line BC is
`vec"r" = (-hat"j" + 3hat"k") + lambda(2hat"i" - 2hat"j" - 4hat"k")`
⇒ `xhat"i" + yhat"j" + zhat"k" = 2lambdahat"i" - (2lambda + 1)hat"j" + lambda(3 - 4lambda)hat"k"`
Comparing both sides, we get
x = `2lambda, y = -(2lambda + 1), z = 3 - 4lambda` .....(1)
Thus, the co-ordinate of L are (2λ, – (2λ + 1), (3 – 4λ),
so that the direction ratios of the line AL are (1 – 2λ), 8 + (2λ + 1), 4 – (3 – 4λ),
i.e. 1 – 2λ, 2λ + 9, 1 + 4λ
Since AL is perpendicular to BC, we have,
(1 – 2λ)(2 – 0) + (2λ + 9)(–3 + 1) + (4λ + 1)(–1 –3) = 0
⇒ λ = `(-5)/6`
The required point is obtained by substituting the value of λ, in (1)
Which is `((-5)/3, 2/3, 19/3)`.
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