SOLUTION: (b) Show that the vectors P = 3i – 2j + k, Q = i – 3j + 5k, and R = 2i + j – 4k form a right angled triangle. (c) Use vectors in (b) above to evaluate P Q and R Q what do these t

Algebra ->  Test -> SOLUTION: (b) Show that the vectors P = 3i – 2j + k, Q = i – 3j + 5k, and R = 2i + j – 4k form a right angled triangle. (c) Use vectors in (b) above to evaluate P Q and R Q what do these t      Log On


   

Question 1083103: (b) Show that the vectors P = 3i – 2j + k, Q = i – 3j + 5k, and R = 2i + j – 4k form a right angled triangle.
(c) Use vectors in (b) above to evaluate P Q and R Q what do these two final vectors represent
.thank you
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of two vectors must equal the other vector,
PQ%2BQR=PR
So then,
PQ=OP-OQ
QR=OQ-OR
PR=OP-OR
where O stands for the origin,
So,

PQ=+%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C-4%29%29
.
.

QR=+%28matrix%281%2C5%2C-1%2C%22%2C%22%2C-4%2C%22%2C%22%2C9%29%29
.
.

PR=+%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29
.
.
And,

PQ%2BQR=%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29
PQ%2BQR=PR
So the two vectors do add up to the third.
Additionally, two of the vectors must be perpendicular to each other in order for that to happen.
So check the dot products,
P%2AQ=3%281%29%2B%28-2%29%28-3%29%2B1%285%29=3%2B6%2B5=14
Q%2AR=1%282%29%2B%28-3%29%281%29%2B5%28-4%29=2-3-20=-21
P%2AR=3%282%29%2B%28-2%29%281%29%2B1%28-4%29=6-2-4=0
Since the dot product is zero, the two vectors are perpendicular (right angle) and together with the result above it proves that the vectors form a right triangle.