Question

Will award medal
Given three points P(1, –1, 0), Q(0, 1, 2) and R(–1, –1, 1), find the distance from the point Q to the line

passing through the points P and R.

Answer 1

@ganeshie8

Answer 2

@iambatman

Answer 3

@Zarkon

Answer 4

Have you done cross product,magnitudes,... with vectors? @pmkat14

Answer 5

yes.
so what im kinda confused on is do i need to do a projection then a component

Answer 6

There are different ways of doing it.
I propose to find the area of triangle PQR, and divide by the base PR, using vectors.

Answer 7

so i have PQ<-1,2,2> PR<-2,0,1>. where PR=v. and a. while PQ=b

Answer 8

then i find the equation of the line as l=(1,-1,0)+t<-2,0,1>

Answer 9

I saw online that i need to take the unit vector of v. but i dont know where that reasoning comes from

Answer 10

That's one way. The area way is easier.
Twice the area of triangle PQR is the area of the parallelogram given by PQ x RQ.
The length of the base is |PR| (magnitude)
so the distance d of Q from PR is just |PQ x RQ| / |PR|
I'll try to draw it.

Answer 11

Examle, |PR|=\(\sqrt{2^2+0^2+(-1)^2}=\sqrt 5\)

Answer 12

Your line is good.

Answer 13

The other way is to find vector QP, then subtract vectorially the projection of QP onto PR, which gives the perpendicular of Q onto PR.
This is ok too!