SOLUTION: Find all values of k such that the triangle in R3 with vertices A = (1, 3, 1), B = (0, 5, 2), C = (k, 3, 1) is a right angled triangle.

Algebra ->  Vectors -> SOLUTION: Find all values of k such that the triangle in R3 with vertices A = (1, 3, 1), B = (0, 5, 2), C = (k, 3, 1) is a right angled triangle.      Log On


   

Question 1043484: Find all values of k such that the triangle in R3 with vertices
A = (1, 3, 1), B = (0, 5, 2), C = (k, 3, 1)
is a right angled triangle.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let vector a = <1,3,1>, b = <0,5,2>, c = <k,3,1>
===> b-a = <-1,2,1>, c-b = <k,-2,-1>, and c-a = <k-1,0,0>.
There are three different cases:
(i) (b-a)*(c-b) = 0 = -k-4-1 = -k-5 ===> -k-5 = 0 ===> k = -5.
(ii) (b-a)*(c-a) = 0 = -(k-1) = 0 ===> k = 1.
(iii) (c-b)*(c-a) = k(k-1) = 0 ===> k = 0, 1.
Therefore, there are three distinct values of k where points A, B, C can form a triangle, namely, 0, 1, and -5.