The Cross Product

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Section 13.4

• The Cross Product

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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.
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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
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THE CROSS PRODUCT AS A DETERMINANT
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THEOREM
The vector a b is orthogonal to both a and b.
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THEOREM
If ? is the angle between a and b (so 0 ? p),
then a b a b sin ?
Corollary Two nonzero vectors a and b are
parallel if and only if a b 0
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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.
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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
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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.
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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)
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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 t (relative to the origin) is defined to
be the cross product of the position and force
vectors. That is, t r F. The magnitude of
the torque is t r F r F sin ?