What is the length PC? How did you solve this?

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1 Answer
Apr 6, 2018

See below.

Explanation:

Calling

vec v_1, vec v_2, vec v_3 the unit vectors with the direction of A-P, B-P and C-P we have

A = P + a vec v_1
B = P + b vec v_2
C = P + c vec v_3

Here

a = 10
b = 6

then

B-A = b vec v_2-a vec v_1
C-B = c vec v_3 - b vec v_2

but B-A and C-B are perpendicular then

<< B-A, C-B >> = << b vec v_2-a vec v_1, c vec v_3 - b vec v_2 >> = 0

b c << vec v_2, vec v_3 >> - b^2 norm(vec v_2)^2-ac << vec v_1, vec v_3 >>+a b << vec v_1, vec v_2 >> =0

and then

c = (b^2-ab << vec v_1, vec v_2>>)/(b << vec v_2, vec v_3 >> - a << vec v_1, vec v_3>>) = 33

with

<< vec v_1, vec v_2 >> =-1/2
<< vec v_2, vec v_3 >> =-1/2
<< vec v_1, vec v_3 >> = -1/2

NOTE

<< cdot, cdot >> represents the scalar product of two vectors.
norm(cdot) represents the norm operation.