Multivariate Calculus - Bruce E. Shapiro

24 LECTURE 3. THE CROSS PRODUCT
be a 2 × 2 square matrix. Then its determinant is given by
det M = |M| =
∣ a b
c d∣ = ad − bc
If M is a 3 × 3 matrix, then
a b c
∣ ∣∣∣
det M = |M| =
d e f
∣g h i ∣ = a e
h
and hence
∣
f
∣∣∣ i ∣ − b d
g
∣
f
∣∣∣ i ∣ + c d
g
det M = a(ei − fh) − b(di − fg) + c(dh − eg) (3.4)
From equation (3.4) we can calculate the following:
⎛ ⎞
det M = ( a b c ) ei − fh
⎝fg − di⎠ (3.5)
dh − eg
Let w = (A, B, C), u = (α, β, γ), and v = (a, b, c), as before. Then comparing
equations (3.3) and (3.5)
⎛ ⎞
w · (u × v) = ( A B C ) cβ − bγ
⎝aγ − cα⎠ (3.6)
bα − aβ
By analogy we can derive the following result.
e
h∣
= A(cβ − bγ) + B(aγ − cα) + C(bα − aβ) (3.7)
= A
∣ β γ
∣ ∣ ∣∣∣ b c∣ α γ
∣∣∣ a c∣ α β
a b∣ (3.8)
⎛ ⎞
A B C
= ⎝α β γ ⎠ (3.9)
a b c
Theorem 3.1 Let v = (a, b, c), u = (α, β, γ), and let i, j, and k be the usual basis
vectors. Then
i j k
u × v =
α β γ
(3.10)
∣a b c∣
Proof.
i j k
α β γ
∣a b c∣
= i
∣ β
b
γ
∣ ∣∣∣ c∣ − j α
a
∣
γ
∣∣∣ c∣ + k α
a
β
b∣
= i(cβ − bγ) − j(cα − aγ) + k(bα − aβ)
= u × v
where the last line follows from equation (3.3).
Revised December 6, 2006. Math 250, Fall 2006