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« Dot Product
Linear Algebra - Vectors: (lesson 3 of 3)

Cross Product

Besides the usual addition of vectors and multiplication of vectors by scalars, there are also two types of multiplication of vectors by other vectors. One type, the dot product, is a scalar product; the result of the dot product of two vectors is a scalar. The other type, called the cross product, is a vector product since it yields another vector rather than a scalar. As with the dot product, the cross product of two vectors contains valuable information about the two vectors themselves.

The cross product of two vectors Cross Productand Cross Product is given by

Cross Product

Although this may seem like a strange definition, its useful properties will soon become evident. There is an easy way to remember the formula for the cross product by using the properties of determinants. Recall that the determinant of a 2x2 matrix is

Cross Product definition

and the determinant of a 3x3 matrix is

Cross Product definition

Notice that we may now write the formula for the cross product as

Cross Product formula

Example 1:

The cross product of the vectors Cross Product exampleand Cross Product example.

Solution:

Cross Product solution

Properties of the Cross Product:

1. The length of the cross product of two vectors is

Properties of the Cross Product

2. Anticommutativity:

Properties of the Cross Product

3. Multiplication by scalars:

Properties of the Cross Product

4. Distributivity:

Properties of the Cross Product

5. The scalar triple product of the vectors a, b, and c:

Properties of the Cross Product

Example 2

Calculate the area of the parallelogram spanned by the vectors a = <3, - 3, 1> and b = <4, 9, 2>.

Solution:

The area is cross product example. Using the above expression for the cross product, we find that the area is cross product example