# If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q - Mathematics and Statistics

Sum

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

#### Solution

Let PQ be the perpendicular drawn from point P(2, 4, 3) to the line joining the points A(1, 2, 4) and B (3, 4, 5).

Let Q divides AB internally in the ratio λ:1

∴ Q ≡ ((3lambda + 1)/(lambda + 1), (4lambda + 2)/(lambda + 1), (5lambda + 4)/(lambda + 1))   .......(i)

Direction ratios of PQ are

(3lambda + 1)/(lambda + 1) - 2, (4lambda + 2)/(lambda + 1) - 4, (5lambda + 4)/(lambda + 1) - 3

i.e., (lambda - 1)/(lambda + 1), (-2)/(lambda + 1), (2lambda + 1)/(lambda + 1)

Now, direction ratios of AB are, 3 – 1, 4 – 2, 5 – 4 i.e., 2, 2, 1.

Since PQ is perpendicular to AB,

2((lambda - 1)/(lambda + 1)) + (2(-2))/(lambda + 1) + 1((2lambda + 1)/(lambda + 1)) = 0

∴ (2lambda - 2 - 4 + 2lambda + 1)/(lambda + 1) = 0

∴ 4λ − 5 = 0

∴ 4λ = 5

∴ λ = 5/4

Putting λ = 5/4 in (i),

Coordinates of Q are,

(3(5/4) + 1)/((5/4) + 1) = 19/9

(4(5/4) + 2)/((5/4) + 1) = 28/9

(5(5/4) + 4)/((5/4) + 1) = 41/9

∴ Q ≡ (19/9, 28/9, 41/9)

Concept: Representation of Vector
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