Page 210 - A Course in Linear Algebra with Applications

194 Chapter Seven: Orthogonality in Vector Spaces

Since this expression cannot be negative, it has a real square
root, which is called the length of X. It is written

T
||X|| = VX X = ^ xl +xl + ... + 2 n .
x
Notice that ||X|| > 0, and \\X\\ = 0 if and only if all the x {
are zero, that is, X = 0. So the only vector of length 0 is the
zero vector. A vector whose length is 1 is called a unit vector.
At this point it is as well to specialize to R 3 where geo-
metrical intuition can be used. Recall that a 3-column vector
3
X in R , with entries xi, X2, £3, is represented by a line seg-
ment in three-dimensional space with arbitrary initial point
(
(CJI, a2, a 3) and endpoint oi+xi, a 2+x 2, 03 + 2:3). Thus the
length of the vector X is just the length of any representing
line segment.
This suggests that we look for a geometrical interpreta-
3
tion of the scalar product of two vectors in R .

Theorem 7.1.1
3
Let X and Y be vectors in R . Then
T
X Y = \\X\\ \\Y\\cos 9.

where 9 is the angle in the interval [0, TT] between line segments
representing X and Y drawn from the same initial point.

Proof
Consider the triangle rule for adding the vectors X and Y — X
in the triangle IAB, as shown in the diagram below.