Page 217 - A Course in Linear Algebra with Applications

7.1: Scalar Products in Euclidean Space 201

Here the determinant is evaluated by expanding along row 1
in the usual manner.
Example 7.1.2

The vector product of X =

1 x 7 = 1 - 1

which becomes on expansion

14'
X x Y = 14i + 8j - 5k = | 8

The importance of the vector product X xY arises from
the fact that it is orthogonal to each of the vectors X and Y;
thus it is represented by a line segment that is normal to the
plane containing line segments corresponding to X and Y, in
case these are not parallel. To see this we can simply form the
scalar product of X x Y in turn with X and Y. For example,

T
X (X xY) =

Since rows 1 and 2 are identical, this is zero by a basic property
of determinants (3.2.2).
In fact the vectors X, Y, X x Y form a right-handed
system in the sense that their directions correspond to the
thumb and first two index fingers of the right hand when held
extended.