Section 13.4

The Cross Product

If , then the cross product of a and b is the vector

THE CROSS PRODUCT

321321 ,,and,, bbbaaa ba

122131132332 ,, babababababa ba

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.

A determinant of order 2 is defined by

DETERMINANTS

bcaddc

ba

21

213

31

312

32

321

321

321

321

cc

bba

cc

bba

cc

bba

ccc

bbb

aaa

A determinant of order 3 can be defined in terms of second order determinants as follows:

THE CROSS PRODUCT AS A DETERMINANT

kji

kji

ba

21

21

31

31

32

32

321

321

bb

aa

bb

aa

bb

aa

bbb

aaa

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.

321

321

321

)(

ccc

bbb

aaa

cba

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)|

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