Calculus 2nd Edition Rogawski

700 C H A P T E R 13 VECTOR GEOMETRY
v × w
w
v
Furthermore, it can be checked directly from Eq. (2) that the determinant does not change
when we replace {u, v, w} by {v, w, u} (or {w, u, v}) . Granting this and using Eq. (6), we
obtain
⎛
v
⎞ ⎛
v × w
⎞
det ⎝ w ⎠ = det ⎝ v ⎠ = (v × w) · (v × w) = ∥v × w∥ 2 > 0
v × w
w
Therefore {v, w, v × w} is right-handed as claimed (Figure 13).
FIGURE 13 Both {v × w, v, w} and
{v, w, v × w} are right-handed.
13.4 SUMMARY
• Determinants of sizes 2 × 2 and 3 × 3:
∣ a ∣
11 a 12 ∣∣∣
= a
a 21 a 11 a 22 − a 12 a 21
22 ∣ a 11 a 12 a 13 ∣∣∣∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣∣ a a 21 a 22 a 23 = a 22 a 23 ∣∣∣ ∣∣∣ a
11 − a 21 a 23 ∣∣∣ ∣∣∣ a
∣
a
a 31 a 32 a 32 a 12 + a 21 a 22 ∣∣∣
33 a 31 a 13 33 a 31 a 32 33
• The cross product of v = ⟨a 1 ,b 1 ,c 1 ⟩ and w = ⟨a 2 ,b 2 ,c 2 ⟩ is the symbolic determinant
v × w =
∣
i j k
a 1 b 1 c 1
a 2 b 2 c 2
∣ ∣∣∣∣∣
=
∣ b ∣ 1 c 1 ∣∣∣ i −
c 2
∣ a ∣ 1 c 1 ∣∣∣ j +
a 2 c 2
∣ a ∣
1 b 1 ∣∣∣
k
b 2
b 2
• The cross product v × w is the unique vector with the following three properties:
(i) v × w is orthogonal to v and w.
(ii) v × w has length ∥v∥∥w∥ sin θ (θ is the angle between v and w, 0 ≤ θ ≤ π).
(iii) {v, w, v × w} is a right-handed system.
• Properties of the cross product:
a 2
k
i
FIGURE 14 Circle for computing the cross
products of the basis vectors.
j
(i) w × v = −v × w
(ii) v × w = 0 if and only if w = λv for some scalar or v = 0
(iii) (λv) × w = v × (λw) = λ(v × w)
(iv) (u + v) × w = u × w + v × w
v × (u + w) = v × u + v × w
• Cross products of standard basis vectors (Figure 14):
i × j = k, j × k = i, k × i = j
• The parallelogram spanned by v and w has area ∥v × w∥.
• Cross-product identity: ∥v × w∥ 2 = ∥v∥ 2 ∥w∥ 2 − (v · w) 2 .
• The vector triple product is defined by u · (v × w). We have
⎛
u · (v × w) = det ⎝
u
v
w
⎞
⎠
• The parallelepiped spanned by u, v, and w has volume |u · (v × w)| .