Isosceles triangle 3D Vectors.

Question

Question " The points A and B have position vectors 10i -23j +10k and pi +14j - 22k respectively, relative to a fixed origin O, where p is a constant.

Given that ΔOAB is isosceles, find three possible positions of point B."

My work "

ΙOBΙ = ΙOAΙ

√( p^2 + 14^2 + (-22)^2 ) = √( 10^2 + (-23)^2 + 10^2 )

√ (p^2 + 680) = 27

p^2 + 680 = 27^2

p^2 + 680 = 729

p^2 = 49

p = ± 7

Don't know how to find the 3rd point. Comparing angles just seems too long for this question. "

Answer "

p = 7 , -7 , (1813)/(20) "

Thanks for the help

Answer 1

I'm lost.

√p^2+680=27 Where did these figures come from?

p+680=27

p=653

OA=OB?