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Vectors 397
MULTIPLICATION OF A VECTOR BY A SCALAR
To multiply a vector by a scalar quantity such as a real number, multiply both the x
and y components of the vector by that scalar. That makes the vector longer or
shorter if the real number is positive, but doesn’t change its direction. If the real
number is negative, the direction of the vector is exactly reversed, and its length may
change as well.
Multiplication by a scalar is commutative. This means that it doesn’t matter whether the
scalar comes before or after the vector in the product. If we have a vector a (x a
,y a
)
and a scalar k, then
ka ak (kx a
,ky a
)
You multiply both the x and y values by the scalar k to get the result.
When you multiply a vector by a scalar, it isn’t true vector multiplication because the
multiplicand and the multiplier aren’t both vectors. You can multiply vectors by other
vectors, however—and things get interesting then! There are two ways to multiply a
vector by another vector. The first way gives you a scalar as the product. The second
(and somewhat more complicated) way gives you a vector as the product. Let’s look at
the first of these two operations now. Later in this chapter, you’ll learn about the second
method.
DOT PRODUCT OF TWO VECTORS
Imagine two vectors a (x a
,y a
) and b (x b
,y b
) such as those shown in Fig. 26-1. If all
four of the variables x a
, x b
, y a
, and y b
are real numbers, then the dot product, also known
as the scalar product and written a · b, of the vectors a and b is a real number and can
be found by using this formula:
a·b x a
x b
y a
y b
The dot product of a and b is called “a dot b” in informal talk.
As an example, let’s find the dot product of the two vectors a (3,–5) and b (2,6).
Use the formula given above.
a · b (3 · 2) (–5 · 6)
6 (–30)
–24
Now let’s see what happens if the order of the dot product is reversed. Does the value
change? If not, the dot product is commutative. Let’s try a general proof. Take the formula