Page 218 - A Course in Linear Algebra with Applications

202 Chapter Seven: Orthogonality in Vector Spaces

Theorem 7.1.3
3
If X and Y are vectors in R ; the vector X xY is orthogonal
to both X and Y, and the three vectors X, Y, X x Y form a
right-handed system.

The length of the vector product, like the the scalar prod-
uct, is a number with geometrical significance.

Theorem 7.1.4
If X and Y are vectors in R 3 and 9 is the angle in the interval
[0,7r] between X and Y, then

\\X xY\\ = \\X\\ \\Y\\sm 9.

Proof
2 2 2
We compute the expression ||X|| ||y|| — \\X x Y\\ , by sub-
2 2 2 2 2
stituting ||X|| = x\ + x\ + x 3, \\Y\\ = y + yj + y and
2 2 2 2
\\X x Y\\ = (x 2y 3 - x 3y 2) + (x 3y x - xxy^) + (x xy 2 - x 2yi) •

After expansion and cancellation of some terms, we find that

2
T
2
||X|| ||F|| 2 - \\X x Y\\ 2 = (x lVl + x 2y 2 + x 3y 3) 2 = (X Y) .
Therefore, by 7.1.1,

2
2
2
2
||X|| ||F|| 2 -\X x Y\ 2 = ||X|| ||y|| cos ^.
2
2
Consequently \\X x Y\\ 2 = \\X\\ 2 ||F|| sin ^. Finally, take
the square root of each side, noting that the positive sign is
correct since sin 9 > 0 in the interval [0, n].
Theorem 7.1.4 provides another geometrical interpreta-
tion of the vector product X x Y. For ||X x Y\\ is simply
the area of the parallelogram IPRQ formed by line segments
representing the vectors X and Y. Indeed the area of this