5.6. GEOMETRIC MEANING OF SCALAR MULTIPLICATION IN R3 63

as claimed. This proves the inequality. ■There are certain properties of the distance which are obvious. Two of them which

follow directly from the definition are

|x−y|= |y−x| ,

|x−y| ≥ 0 and equals 0 only if y = x.

The third fundamental property of distance is known as the triangle inequality. Recall thatin any triangle the sum of the lengths of two sides is always at least as large as the thirdside. The following corollary is equivalent to this simple statement.

Corollary 5.5.5 Let x,y be points of Rn. Then

|x+y| ≤ |x|+ |y| .

Proof: Using the Cauchy Schwarz inequality, Lemma 5.5.4,

|x+y|2 ≡n

∑i=1

(xi + yi)2

=n

∑i=1

x2i +2

n

∑i=1

xiyi +n

∑i=1

y2i

≤ |x|2 +2 |x| |y|+ |y|2

= (|x|+ |y|)2

and so upon taking square roots of both sides,

|x+y| ≤ |x|+ |y|

■

5.6 Geometric Meaning Of Scalar Multiplication In R3

As discussed earlier, x = (x1,x2,x3) determines a vector. You draw the line from 0 tox placing the point of the vector on x. What is the length of this vector? The lengthof this vector is defined to equal |x| as in Definition 5.5.1. 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 is defined 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 the length of the vector. What aboutthe direction? You should convince yourself by drawing a picture that if α is negative, itcauses the resulting vector to point in the opposite direction while if α > 0 it preserves thedirection the vector points. One way to see this is to first observe that if α ̸= 1, then x andαx are both points on the same line. Note that there is no change in this when you replaceR3 with Rn.