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ê × ê = ê ; ê × ê = ê ; ê × ê = ê
1 2 3 2 3 1 3 1 2

Pb. 7.9 Ask your instructor to show you how the Right Hand rule is used to
determine the direction of a vector equal to the cross product of two other
vectors.

7.5.2 Geometric Interpretation of the Cross Product

As noted in Pb. 7.7a, the cross product is a vector that is perpendicular to its
two constituents. This determines the resultant vector’s direction. To deter-
r
mine its magnitude, consider the Lagrange Identity. If the angle between u
r
and is θ, then:v
r
r 2
r 2
u v× r 2 = u r 2 v − u r 2 v cos ( )θ (7.50)
2
and

r r r r
uv× = u v sin( )θ (7.51)

that is, the magnitude of the cross product of two vectors is the area of the
parallelogram formed by these vectors.

7.5.3 Scalar Triple Product
r r r r r r
DEFINITION If uv, ,and r r are vectors in 3-D, then uv w⋅( × ) is called the
w
r
scalar triple product of uv,,and w.
PROPERTY

r r r r r r r r r
uv w⋅( × ) = v w u⋅( × ) = wu v⋅( × ) (7.52)

This property can be trivially proven by writing out the components expan-
sions of the three quantities.

7.5.3.1 Geometric Interpretation of the Scalar Triple Product
r r r
If the vectors’ uv, ,and w original points are brought to the same origin, these
three vectors define a parallelepiped. The absolute value of the scalar triple
product can then be interpreted as the volume of this parallelepiped. We have
r
r
r
r
shown earlier that vw× is a vector that is perpendicular to both v and w ,
© 2001 by CRC Press LLC