Section 11.3: The Cross Product
Definition: The cross product of two vectors ~v = hv1 , v2 , v3 i and w
~ = hw1 , w2 , w3 i is the
vector given by
~v × w
~ = hv2 w3 − v3 w2 , v3 w1 − v1 w3 , v1 w2 − v2 w1 i.
Note: The dot product of two vectors is a scalar, but the cross product is a vector. An easy
way to remember this formula is to use the notation of determinants.
Definition: A determinant of order 2 is given by
a b c d = ad − bc.
A determinant of order 3
a1 a2 a3
b1 b2 b3
c1 c2 c3
is given by
b1 b2 b1 b3 b2 b3 = a1 c 2 c 3 − a2 c 1 c 3 + a3 c 1 c 2 .
Example: Evaluate the determinant
Using the definition,
1 2 4 3 6 5 = 1 6 5
1 −2
0 1 −2 −2 3 5
0 −2
1 2 4
3 6 5
0 1 −2
.
+4 3 6
0 1
= 1(−12 − 5) − 2(−6 − 0) +4(3 − 0) = 7.
Note: The cross product of ~v = hv1 , v2 , v3 i and w
~ = hw1 , w2 , w3 i is given by the determinant
~i ~j ~k ~v × w
~ = v1 v2 v3 = hv2 w3 − v3 w2 , v3 w1 − v1 w3 , v1 w2 − v2 w1 i.
w1 w2 w3 Example: Find the cross product of ~v = h1, 1, 3i and w
~ = h−2, −1, −5i.
~v × w
~ = ~k
~i
~j
1
1
3
−2 −1 −5
= h−5 + 3, −6 + 5, −1 + 2i = h−2, −1, 1i.
Theorem: (Orthogonality of the Cross Product)
If ~v and w
~ are nonzero vectors in R3 that are not scalar multiples of one another, then ~v × w
~
is orthogonal to both ~v and w.
~
~ is determined by the right-hand rule.
Note: The direction of ~v × w
𝑣×𝑤
𝑤
𝑣
Figure 1: The cross product of ~v and w
~ illustrating the right-hand rule.
Example: Find a unit vector orthogonal to both ~v = h1, 2, 1i and w
~ = h0, 1, 3i.
The cross product is
~v × w
~ = ~i
1
0
~j
2
1
~k
1
3
= h6 − 1, 0 − 3, 1 − 0i = h5, −3, 1i.
The magnitude of the cross product is
||~v × w||
~ =
√
25 + 9 + 1 =
√
35.
Thus, a unit vector orthogonal to ~v and w
~ is
~v × w
~
5
3
1
~u =
= √ , −√ , √
.
||~v × w||
~
35
35 35
Example: Find a vector orthogonal to the plane passing through the points A = (1, 2, 1),
B = (3, 5, 0), and C = (5, 4, −2).
The plane contains the vectors
−→
AB = h3 − 1, 5 − 2, 0 − 1i = h2, 3, −1i,
−→
AC = h5 − 1, 4 − 2, −2 − 1i = h4, 2, −3i.
The cross product is orthogonal to the plane
~i ~j ~k −→
−→
AB × AC = 2 3 −1 = h−7, 2, −8i.
4 2 −3 Another vector orthogonal to the plane is −(~v × w)
~ = h7, −2, 8i.
Theorem: (Magnitude of the Cross Product)
If ~v and w
~ are nonzero vectors in R3 and θ is the angle between ~v and w
~ (0 ≤ θ ≤ π), then
||~v × w||
~ = ||~v ||||w||
~ sin θ.
Note: The area of the parallelogram defined by ~v and w
~ is given by ||~v × w||.
~
𝑤
sin 𝜃 =
ℎ
ℎ
𝑤
ℎ = 𝑤 sin 𝜃
𝜃
𝑣
𝐴𝑟𝑒𝑎 = 𝐵𝑎𝑠𝑒 × Height = 𝑣
𝑤 sin 𝜃 = 𝑣 × 𝑤
Figure 2: Illustration of ||~v × w||
~ as the area of a parallelogram.
Example: Find the area of the parallelogram defined by ~v = h1, 1, −2i and w
~ = h2, −1, −1i.
The cross product is
~v × w
~ = ~k ~i ~j
1 1 −2 = h−3, −3, −3i.
2 −1 −1 Then the area of the parallelogram is
||~v × w||
~ =
√
√
9 + 9 + 9 = 3 3.
Definition: If ~u, ~v , and w
~ are nonzero vectors that do not lie in the same plane, then the
scalar triple product is given by
(~u × ~v ) · w.
~
Theorem: (Volume of a Parallelepiped)
The volume of the parallelepiped defined by the vectors ~u, ~v , and w
~ is given by
V = |(~u × ~v ) · w|.
~
𝑤
𝑣
𝑢
Figure 3: Parallelepiped defined by the vectors ~u, ~v , and w.
~
Example: Find the volume of the parallelepiped defined by the vectors ~u = h2, 1, −1i, ~v =
h3, 0, 1i, w
~ = h0, 1, 1i.
The cross product of ~u and ~v is
~i ~j ~k ~u × ~v = 2 1 −1 = h1, −5, −3i.
3 0 1 The volume of the parallelepiped is
V = |(~u × ~v ) · w|
~ = |1(0) − 5(1) − 3(1)| = 8.