Math 152 Class Notes
December 3, 2015
11.3 The Cross Product
In this section, we study the cross product of two vectors which, unlike the dot product,
results in a vector.
Denition. The cross product of two vectors a and b is the vector
a × b = (|a||b| sin θ) n
where θ is the angle between the vectors a and b, 0 ≤ θ ≤ π and n is a unit vector
perpendicular to both a and b and whose direction is given by the right-hand rule:
If the ngers of your right hand curl through the angle θ from a to b, then your thumb
points in the direction of n.
Example 1. Find the cross products i × i, j × j, k × k, i × j, j × i, j × k, k × j, k × i,
and i × k.
Properties of the cross product
1. a × b is orthogonal to both a and b.
2. |a × b| = |a||b| sin θ
3. Two nonzero vectors a and b are parallel if and only if a × b = 0.
4. a × b = −b × a
5. (ka) × b = k(a × b) = a × (kb)
6. c × (a + b) = c × a + c × b
7. (a + b) × c = a × c + b × c
By Properties 6 and 7 (the Vector Distributive Laws), we can compute the cross
product of two vectors a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k in terms of their
components.
a × b = (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k
The cross product can be dened using determinant.
A determinant of order 2 is dened by
a b c d = ad − bc
A determinant of order 3 is dened by
a1 a2 a3 b2 b3 b1 b3 b1 b2 b1 b2 b3 =a1 c2 c3 − a2 c1 c3 + a3 c1 c2 c1 c2 c3 =a1 (b2 c3 − b3 c2 ) − a2 (b1 c3 − b3 c1 ) + a3 (b1 c2 − b2 c1 )
The cross product of a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k is
i j k a2 a3 a1 a3 a1 a2 i − a × b = a1 a2 a3 = b1 b3 j + b1 b2 k
b
b
2
3
b1 b2 b3 =(a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k
Example 1. Find the cross product of a = h1, 2, −1i and b = h3, −1, 7i
Example 2. Find a vector perpendicular to the plane that passes through the points
A(2, 3, −2), B(1, −1, 0) and C(2, 0, 3)
In geometry, we have the following interpretation of the magnitude of a cross product.
The length of the cross product a × b is equal to the area of the parallelogram
determined by a and b.
Example 3. Given the points A(5, 5, 1), B(3, 3, 2) and C(1, 4, 4), nd the area of
4ABC .
For three vectors a, b and c, we dene their scalar triple product as
a · (b × c)
The geometric signicance of the scalar triple product can be seen by considering the
parallelepiped determined by the vectors a, b and c.
The volume of the parallelepiped determined by the vectors a, b and c is the
magnitude of their scalar triple product:
V = |a · (b × c)|
Example 4. The volume of the parallelepiped whose corner is formed by the vectors
a = h2, −1, 4i, b = h1, −3, 0i and c = h3, 1, −2i.
Example 5. Given the points A(1, 0, 1), B(2, 3, 0), C(−1, 1, 4) and D(0, 3, 2), nd the
−→ −→
−−→
volume of the parallelepiped with adjacent edges AB , AC and AD.