2.5. GEOMETRIC MEANING OF SCALAR MULTIPLICATION 23

Example 2.4.4 Here is a picture of two vectors, u and v.

u

v

Sketch a picture of u+v,u−v.

First here is a picture of u+v. You first draw u and then at the point of u you place thetail of v as shown. Then u+v is the vector which results which is drawn in the followingpretty picture.

uv

u+v

Next consider u−v. This means u+(−v) . From the above geometric description ofvector addition,−v is the vector which has the same length but which points in the oppositedirection to v. Here is a picture.

u

−v

u+(−v)

2.5 Geometric Meaning Of Scalar MultiplicationAs discussed earlier, x=(x1,x2,x3) determines a vector. You draw the line from 0 to x plac-ing the point of the vector on x. What is the length of this vector? The length of this vector

is defined to equal |x| as in Definition 2.4.1. Thus the length of x equals√

x21 + x2

2 + x23.

When you multiply x by a scalar α, you get (αx1,αx2,αx3) and the length of this vector isdefined as √(

(αx1)2 +(αx2)

2 +(αx3)2)= |α|

√x2

1 + x22 + x2

3.

Thus the following holds.|αx|= |α| |x| .

In other words, multiplication by a scalar magnifies or shrinks the length of the vector.What about the direction? You should convince yourself by drawing a picture that if α isnegative, it causes the resulting vector to point in the opposite direction while if α > 0 itpreserves the direction the vector points.