Kenneth Kuttler

68 CHAPTER 5. FUNDAMENTALS

a vector, the vector which is meant, is this position vector just described. Another termassociated with this is standard position. A vector is in standard position if the tail isplaced at the origin.

It is customary to identify the point in Rn with its position vector.The magnitude of a vector determined by a directed line segment−→pq is just the distance

between the point p and the point q. By the distance formula this equals(n

∑k=1

(qk− pk)2

)1/2

= |p−q|

and for v any vector in Rn the magnitude of v equals(∑

nk=1 v2

)1/2= |v|.

Example 5.8.3 Consider the vector v ≡ (1,2,3) in Rn. Find |v| .

First, the vector is the directed line segment (arrow) which has its base at 0 ≡ (0,0,0)and its point at (1,2,3) . Therefore,

|v|=√

12 +22 +32 =√

14.

What is the geometric significance of scalar multiplication? If a represents the vectorv in the sense that when it is slid to place its tail at the origin, the element of Rn at its pointis a, what is rv?

|rv|=

∑k=1

(rai)2

)1/2

∑k=1

r2 (ai)2

)1/2

=(r2)1/2

∑k=1

a2i

)1/2

= |r| |v| .

Thus the magnitude of rv equals |r| times the magnitude of v. If r is positive, then thevector represented by rv has the same direction as the vector v because multiplying by thescalar r, only has the effect of scaling all the distances. Thus the unit distance along anycoordinate axis now has length r and in this re-scaled system the vector is represented bya. If r < 0 similar considerations apply except in this case all the ai also change sign. Fromnow on, a will be referred to as a vector instead of an element of Rn representing a vectoras just described. The following picture illustrates the effect of scalar multiplication.

v 2v −2v

Note there are n special vectors which point along the coordinate axes. These are

ei ≡ (0, · · · ,0,1,0, · · · ,0)

68 CHAPTER 5. FUNDAMENTALSa vector, the vector which is meant, is this position vector just described. Another termassociated with this is standard position. A vector is in standard position if the tail isplaced at the origin.It is customary to identify the point in R” with its position vector.The magnitude of a vector determined by a directed line segment pq is just the distancebetween the point p and the point q. By the distance formula this equalsh 1/2(3 n-ne] =|p-q|k=1and for v any vector in R" the magnitude of v equals (Lf_; vz) 2 |v.Example 5.8.3 Consider the vector v = (1,2,3) in R”. Find |v|.First, the vector is the directed line segment (arrow) which has its base at 0 = (0,0,0)and its point at (1,2,3). Therefore,jv) = V 174274 32 = v14.What is the geometric significance of scalar multiplication? If @ represents the vectorv in the sense that when it is slid to place its tail at the origin, the element of R” at its pointis a, what is rv?Ch 1/2 Ch 1/2|rv| = (x i?) = e r 0?)k=1 k=lCh 1/2=(7)"" (E4) = |r| 0).k=1Thus the magnitude of rv equals |r| times the magnitude of v. If r is positive, then thevector represented by rv has the same direction as the vector v because multiplying by thescalar r, only has the effect of scaling all the distances. Thus the unit distance along anycoordinate axis now has length r and in this re-scaled system the vector is represented bya. If r < 0 similar considerations apply except in this case all the a; also change sign. Fromnow on, a will be referred to as a vector instead of an element of R” representing a vectoras just described. The following picture illustrates the effect of scalar multiplication.fy)Note there are 1 special vectors which point along the coordinate axes. These aree; = (0,-+ ,0,1,0,--- ,0)