section 13.4 the cross product. if, then the cross product of a and b is the vector the cross...
TRANSCRIPT
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