Question

The coordinates of the foot of the perpendicular drawn from the point A(1, 0, 3) to the join of the points B(4, 7, 1) and C(3, 5, 3) are

Options

• `(5/3, 7/3, 17/3)`

• `(5, 7, 17)`

• `(5/7, 7/3, 17/3)`

• `(5/3, 7/3, 17/3)`

Answer 1

`(5/3, 7/3, 17/3)`

Explanation:

Let D be the foot of the perpendicular and let it divide BC in the ratio λ : 1. Then, the coordinates of D are

`((3lambda + 4)/(lambda + 1), (5lambda + 7)/(lambda + 1), (3lambda + 1)/(lambda + 1))`

Now, AD ⊥ BC

⇒ AD ⋅ BC = 0

⇒ `- (2lambda + 3) - 2(5lambda + 7) - 4` = 0

⇒ `lambda = - 7/4`

So, the coordinates of D are `(5/3, 7/3, 17/3)`