Use the cross product to find the area of the triangle with vertices:

Question

P=(1,1,5)
Q=(3,4,3)
R=(1,5,7)

Answer 1

The area of the triangle is half of the area of a parallelogram having P, Q and R as three of its four vertices, and assume the fourth to be Q'.

The area of the parallogram PQRQ' is the magnitude of the cross product of the two vectors QP and QR.

Using P=(1,1,5), Q=(3,4,3), R=(1,5,7),
QP = (-2,-3,2) and
QR = (-2,1,4)

QPxQR=
|i j k|
|-2 -3 2|
|-2 1 4|
= (-14,4,-8)
Magnitude of the cross product
= |QPxQR|
= |(-14,4,-8)|
= √276

Area of the triangle
= (1/2)√276
=√69