(a) Draw a picture of a triangle with two sides a and b. Write down a formula for the height of the triangle in terms

crossproduct
height-of-triangle

#1

(a) Draw a picture of a triangle with two sides a and b. Write down a formula for the
height of the triangle in terms of the crossproduct, and deduce a formula for the area of the triangle.
(b) Find the area of a triangle with corners P(0,1,0), Q(1,2,−1), R(2,2,2).

Answer:

height of triangle = Area of parallelogram / base of parallelogram = a x b / [a]

Coordinate vectors; PQ=(1,1,-1) PR=(2,1,2) let a = PQ and b = PR

a x b = ( a2b3-a3b2 , a3b1-a1b3 , a1b2-a2b1)

= (3,-4,1)

area of triangle = 1/2 * [a x b] = 0.5 x (3^2 + -4^2 + 1^2)^0.5 = 2.55