Find the foot of the perpendicular drawn from the point to the line Also, find the length of the perpendicular.

Question

Find the foot of the perpendicular drawn from the point  to the line  Also, find the length of the perpendicular.

Philoid · Accepted Answer

Given: - Point with position vector and equation of line

Let, PQ be the perpendicular drawn from P () to given line whose endpoint/ foot is Q point.

Q is on line

⇒ is the position vector of Q

Hence,

⇒

Since, PQ is perpendicular on line

Therefore, their Dot product is zero

Compare given line equation with

⇒

⇒ 14λ – 14 = 0

⇒ λ = 1

Hence Position vector of Q by putting the value of λ

⇒

⇒ ; Foot of perpendicular

and Distance PQ, putting the value of λ in PQ vector equation, we get

⇒

Now, Magnitude of PQ

We know that,

; where x,y,z are coefficient of vector

Hence

⇒

⇒ units