Homework Solution Section Explorations Cross Product

6. Prove that (a)u◦(v×w) = (u×v)◦w; (b)u×(v×w) = (u◦w)v(u◦v)w; (c)‖u×v‖2 = ‖u‖2‖vecv‖2−(u ◦ v)2.

Proof: Follow from the defintions of Cross-product, dot protduct, and norma.

8. Let u,v be vectors in <3 and let θ be the angle between them.

(a) We know from 6(c) that‖u× v‖2 = ‖u‖2‖v‖2 − (u ◦ v)2.

But from the definition of the angle between two vectors, we have

cos θ =u ◦ v‖u‖‖v‖

⇒ u ◦ v = cos θ(‖u‖‖v‖)

So,‖u× v‖2 = ‖u‖2‖v‖2 − (cos θ(‖u‖‖v‖)2 = ‖u‖2‖v‖2

[1− cos2 θ

]= ‖u‖2‖v‖2(sin2 θ)

Hence, ‖u× v‖ = ‖u‖‖v‖(sin θ)

(b) The area of the triangle is (1/2)(height)(altitude). So, for the given triangle

Area =1

2(‖u‖)(‖v‖ sin θ)

From part (a), we know that this is the same as

Area =1

2‖u× v‖

(c) For the given points A = (1, 2, 1), B = (2, 1, 0), C = (5,−1, 3), we define u =−→AB =

[1−1−1

],v =

−→AC =[

4−32

]. Then, from part (b), the area is

1

2

√62.