Page 168 - Advanced engineering mathematics

148 CHAPTER 6 Vectors and Vector Spaces

z
z

(x,y,z)
<x,y,z>

y
y

x
x
FIGURE 6.1 Vector < x, y, z > from the
FIGURE 6.2 Arrow representations
origin to the point (x, y, z).
of the same vector.
z

<x,y,z>
<–x,–y,–z>

y

x
FIGURE 6.3 < −x,−y,−z > is opposite
< x, y, z >.

The length (also called the magnitude or norm) of a vector F =< x, y, z > is the scalar

2
2
2
F = x + y + z .
This is the distance from the origin to the point (x, y, z) and also the length of any arrow repre-
√
senting the vector < x, y, z >. For example, the norm of G =< −1,4,2 > is G = 21, which
is the distance from the origin to the point (−1,4,2).

Multiply a vector F =< a,b,c > by a scalar α by multiplying each component of F by α.
This produces a new vector denoted αF:

αF =<αa,αb,αc >.

Then

αF = |α| F ,

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October 14, 2010 14:21 THM/NEIL Page-148 27410_06_ch06_p145-186