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3.4 In many applications of vectors to problems in geometry, physics, and
engineering, it is of interest to construct a vector in 3-space that is perpendicular
CROSS PRODUCT to two given vectors. In this section we shall show how to do this.
Cross Product of Vectors
Recall from Section 3.3 that the dot product of two vectors in 2-space or 3-space produces a scalar. We will now define a type of
vector multiplication that produces a vector as the product but that is applicable only in 3-space.
DEFINITION are vectors in 3-space, then the cross product is the vector defined by
If and (1)
or, in determinant notation,
Remark Instead of memorizing 1, you can obtain the components of as follows:
Form the 2 × 3 matrix whose first row contains the components of u and whose second row contains the
components of v.
To find the first component of , delete the first column and take the determinant; to find the second component, delete
the second column and take the negative of the determinant; and to find the third component, delete the third column and take
the determinant.
EXAMPLE 1 Calculating a Cross Product
Find , where and .
Solution
From either 1 or the mnemonic in the preceding remark, we have
