Page 169 - Advanced engineering mathematics

6.1 Vectors in the Plane and 3-Space 149

because

2
2
αF = (αa) + (αb) + (αc) 2
√

2
2
= (α )(a + b + c ) =|α| a + b + c 2
2
2
2
2
=|α| F .
This means that the length of αF is |α| times the length of F. We may therefore think of multipli-
cation of a vector by a scalar as a scaling (stretching or shrinking) operation. In particular, take
the following cases:
• If α> 1, then αF is longer than F and in the same direction.
• If 0 <α < 1, then αF is shorter than F and in the same direction.
• If −1 <α < 0 then αF is shorter than F and in the opposite direction.
• If α< −1 then αF is longer than F and in the opposite direction.
• If α =−1 then αF has the same length as F, and exactly opposite the direction.
1
For example, F is a vector having the direction of F and half the length of F, while 2F
2
1
has the direction of F and length twice that of F, and − F has direction opposite that of F
2
and half the length.
• If α = 0, then αF =< 0,0,0 >, which we call the zero vector and denote O. This is the only
vector with zero length and no direction, since it cannot be represented by an arrow.
Consistent with these interpretations of αF, we define two vectors F and G to be parallel if
each is a nonzero scalar multiple of the other. Parallel vectors may differ in length and even be
in opposite directions, but the straight lines through arrows representing them are parallel lines
in 3-space.

We add two vectors by adding their respective components:
If F =< a 1 ,a 2 ,a 3 > and G =< b 1 ,b 2 ,b 3 >, then
F + G =< a 1 + a 2 ,b 1 + b 2 ,c 1 + c 2 >.

Vector addition and multiplication by scalars have the following properties:

1. F + G = G + F. (commutativity)
2. F + (G + H) = (F + G) + H. (associativity)
3. F + O = F.
4. α(F + G) = αF + αG.
5. (αβ)F = α(βF).
6. (α + β)F = αF + βF.
It is sometimes useful to represent vector addition by the parallelogram law.If F and G are
drawn as arrows from the same point, they form two sides of a parallelogram. The arrow along
the diagonal of this parallelogram represents the sum F + G (Figure 6.4). Because any arrows
having the same lengths and direction represent the same vector, we can also draw the arrows in
F + G (as in Figure 6.5) with G drawn from the tip of F. This still puts F + G along the diagonal
of the parallelogram.
The triangle of Figure 6.5 also suggests an important inequality involving vector sums and
lengths. This triangle has sides of length F , G , and F + G . Because the sum of the

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October 14, 2010 14:21 THM/NEIL Page-149 27410_06_ch06_p145-186