PLEASE HELP

Question

Q1) A triangle is said to be isosceles when two of its edges are the same length. 

(i) Let p=(2,4,−4), q=(5,1,2) and r=(4,3,−1) be the position vectors of the points P,Q and R. Is the triangle PQR isosceles? 
Yes or No?

(ii) Find the centroid G of the triangle PQR. 

(iii) Is the vector RG perpendicular to PQ? 
Yes or No?

 

Could you also show working in terms of PQR and simplified, i am very confused. Thank you!

Answer 1

(ii)

Let p=(2,4,−4), q=(5,1,2) and r=(4,3,−1)

Calculate the centroid:

G =((x₁+x₂+x₃ )/3,(y_₁+y_₂+y₃)/3,(z_₁+z_₂+z₃)/3)

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

    =(11/3,8/3,-1)

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(iii)

                =-1+1+0

                =0

Therefore, PQ and RG are perpendicular.