Page 171 - Advanced engineering mathematics

6.1 Vectors in the Plane and 3-Space 151

We call ai+bj+ck the standard representation of F. When a component of a vector is zero,
we usually just omit this term in the standard representation. For example, we would usually write
F =< −8,0,3 > as −8i + 3k instead of −8i + 0j + 3k.
If a vector is represented by an arrow in the x, y-plane, we often omit the third coordinate
and use i =< 1,0 > and j =< 0,1 >. For example, the vector V from the origin to the point
<2,−6,0> can be represented as an arrow from the origin to the point (2,−6) in the x, y-plane
and can be written in standard form as
V = 2i − 6j

where i =< 1,0 > and j =< 0,1 >.
It is often useful use to know the components of the vector V represented by the arrow from
one point to another, say from P 0 = (x 0 , y 0 , z 0 ) to P 1 : (x 1 , y 1 , z 1 ). Denote

G = x 0 i + y 0 j + z 0 k and F = x 1 i + y 1 j + z 1 k.
By the parallelogram law in Figure 6.7, the vector V we want satisfies
G + V = F.
Therefore,

V = F − G = (x 1 − x 0 )i + (y 1 − y 0 )j + (z 1 − z 0 )k.
For example, the vector represented by the arrow from (−1,6,3) to (9,−1,−7) if 10i−7j−10k.
Using this idea, we can find a vector of any length in any given direction. For example,
suppose we want a vector of length 7 in the direction from (−1,6,5) to (−8,4,9).
The strategy is to first find a unit vector in the given direction, then multiply it by 7 to obtain
a vector of length 7 in that direction. The vector V =−7i − 2j + 4k is in the direction from
√
(−1,6,5) to (−8,4,9). Since V = 69, a unit vector in this direction is
1 1
F = V = √ V.
V 69
Then
7
7F = √ (−7i − 2j + 4k)
69
has length 7 and is in the direction from (−1,6,5) to (−8,4,9).

z
(x ,y ,z )
0
0 0
V = F – G
G

(x ,y ,z )
1 1
1
F
y

x
FIGURE 6.7 Vector from (x 0 , y 0 , z 0 ) to
(x 1 , y 1 , z 1 ).

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:21 THM/NEIL Page-151 27410_06_ch06_p145-186