1. Vector Operations
DEFINITION Let u = (1, y1) and v = (x2. y2) be two vectors in the plane. The sum of the
vectors u and v is the vectoor
and is denoted by u +v. Thus vectors are added by adding their components.
EXAMPLE9 Let u =
(1,2) and v =
=(3, -4). Then
u+V = (1+3, 2+ (-4)) = (4, -2).
We can interpret vector addition geometrically as follows. In Figure 4.14,
the vector from (x1, y1) to (X + X2, y + y2) is also v. Thus the vector with
tail O and head (x1 +X2, yi + y2) is u+v.
Figure4.14
Vector addition
i+2 (XtX2y+y2)
u +v
(X2, y2)
y2
O X2 1tX2
X
u + V
Figure 4.15 A
Vector addition
We can also describe u + v as the diagonal of the parallelogram detined
by u and v, as shown in Figure 4.15.
Finally, observe that vector addition is a special case of matrix addition.
2. EXAMPLE10 Ifu and v are as in Example 9, then u +v is as shown in Figure 4.16.
Figure4.16
(1.2)
X
+
u
( 4 . - 2)
(3,-4)
of u by c is the vector (cx|, cy1). Thus the scalar multiple cu of u by c is
obtained by multiplying each component of u by c.
Ifc> 0, then cuis in the same direction as u, whereas if d < 0, then du
is in the opposite direction (Figure 4.17).
DEFINITION Ifu =
(x1. y1) and c is a scalar (a real number), then the scalar multiple cu
Figure4.17
Scalar multiplication
2u
-2u
EXAMPLE11 Ifc =
=
2, d =
=-3, and u =
(1, -2), then
cu =
2(1, -2) =
(2, -4) and du =
-3(1,-2) =
(-3,6).
which are shown in Figure 4. 18.
The vector (0, 0) is called the zero vector and is denoted by 0. If u is any
vector, it follows that (Exercise T.2)
(4)
u+0= u.
We can also show (Exercise T.3) that
(5)
ut-Du=0.
and we write (1)u as -u and eall it the negative of u. Moreover, we wr
u+-1 v as uv and call it the differenceof u and v. The vectoru-
shown in Figure 4.19(a).
Observe that while vector addition gives one diagonal of a
parallelogran
vector suburaction gives the other diagonal. See Figure 4.19(6).
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