Page 216 - A Course in Linear Algebra with Applications

200 Chapter Seven: Orthogonality in Vector Spaces

3
Vector products in R
In addition to the scalar product, there is another well-
known construction in R 3 called the vector product. This is
defined in the following manner.
Suppose that

Xi \ Vi
X = x 2 and Y = V2
X J \V3
3
3
are two vectors in R . Then the vector product of X and Y
X x Y

is defined to be the vector
( x 2y 3 - £32/2 \

X\y 2 - x 2y x /
Notice that each entry of this vector is a 2 x 2 determinant.
Because of this, the vector product is best written as a
3 x 3 determinant. Following a commonly used notation, let
3
us write i, j , k for the vectors of the standard basis of R .
Thus
0
.-(i),j=(?)„d =( o
k

Then the vector product X xY can be expressed in the form
X x Y = (x 2y 3 - x 3y 2)i + (x 3yi - x xy 3)} + {xxy 2 - Z22/i)k.

This expression is a row expansion of the 3 x 3 determinant

X xY =