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Unformatted text preview: gure 8(a). Alternatively, since v
u, the vector u v, when added to v, gives u. So we could construct u v as in Figure 8(b) by
means of the Triangle Law. v u
u FIGURE 8 Drawing u-v (a) (b) EXAMPLE 2 If a and b are the vectors shown in Figure 9, draw a 2 b. SOLUTION We ﬁrst draw the vector 2 b pointing in the direction opposite to b and twice
as long. We place it with its tail at the tip of a and then use the Triangle Law to draw
2 b as in Figure 10.
a _2 b
b a-2 b FIGURE 9 FIGURE 10 Components
y For some purposes it’s best to introduce a coordinate system and treat vectors algebraically. If we place the initial point of a vector a at the origin of a rectangular coordinate
system, then the terminal point of a has coordinates of the form a1, a2 or a1, a2, a3 ,
depending on whether our coordinate system is two- or three-dimensional (see Figure 11).
These coordinates are called the components of a and we write (a¡, a™) a
O x a=k a¡, a™ l a z
(a¡, a™, a£) a
y x a 1, a 2 or a a 1, a 2 , a 3 We use the notation a1, a2 for the ordered pair that refers to a vector so as not to confuse
it with the ordered pair a1, a2 that refers to a point in the plane.
For instance, the vectors shown in Figure 12 are all equivalent to the vector
3, 2 whose terminal point is P 3, 2 . What they have in common is that the terminal point is reached from the initial point by a displacement of three units to the right
and two upward. We can think of all these geometric vectors as representations of the
y a=k a¡, a™, a£ l z (4, 5) FIGURE 11
(1, 3) position
vector of P P(3, 2) P(a¡, a™, a£)
0 x O x F IGURE 12 Representations of the vector a=k3, 2l B(x+a¡, y+a™, z+a£) A(x, y, z) FIGURE 13
Representations of a=ka¡, a™, a£l y 5E-13(pp 828-837) 1/18/06 11:11 AM Page 837 S ECTION 13.2 VECTORS ❙❙❙❙ 837 l
3, 2 . The particular representation OP from the origin to the point
algebraic vector a
P 3, 2 is called the position vector of the point P.
a1, a2, a3 is the position vector of the...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
- Fall '09