1. Section 13.4
The Cross Product

2. THE CROSS PRODUCT If , then the cross product of a and b is the vector
NOTES:
1. The cross product is also called the vector product.
2. The cross product a × b is defined only when a and b are three-dimensional vectors.

3. DETERMINANTS A determinant of order 2 is defined by
A determinant of order 3 can be defined in terms of second order determinants as follows:

4. THE CROSS PRODUCT AS A DETERMINANT

5. THEOREM
The vector a × b is orthogonal to both a and b.

6. THEOREM If θ is the angle between a and b (so 0 ≤ θ ≤ π), then
|a × b| = |a| |b| sin θ
Corollary: Two nonzero vectors a and b are parallel if and only if
a × b = 0

7. A GEOMETRIC INTERPRETATION OF THE CROSS PRODUCT
The length of the cross product a × b is equal to the area of the parallelogram determined by a and b.

8. PROPERTIES OF THE CROSS PRODUCT
If a, b, and c are vectors and c is a scalar, then
1. a × b = −(b × a)
2. (ca) × b = c(a × b) = a × (cb)
3. a × (b + c) = a × b + a × c
4. (a + b) × c = a × c + b × c
5. a ∙ (b × c) = (a × b) ∙ c
6. a × (b × c) = (a ∙ c)b − (a ∙ b)c

9. SCALAR TRIPLE PRODUCT
The product a ∙ (b × c) is called the scalar triple product of vectors a, b, and c.
NOTE: The scalar triple product can be computed as a determinant.

10. GEOMETRIC INTERPRETATION OF THE SCALAR TRIPLE PRODUCT
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)|

11. |τ| = |r × F| = |r| |F| sin θ
TORQUE
Consider a force F acting on a rigid body at a point given by the position vector r. (For example, tightening a bolt with a wrench.) The torque τ (relative to the origin) is defined to be the cross product of the position and force vectors. That is, τ = r × F.
The magnitude of the torque is
|τ| = |r × F| = |r| |F| sin θ