Section 13.4 The Cross Product

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.

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:

THE CROSS PRODUCT AS A DETERMINANT

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

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

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.

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

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.

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)|

|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