Kenneth Kuttler

13.6. GEOMETRIC MEANING OF SCALAR MULTIPLICATION IN R3 281

13.6 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 length ofthis vector is defined to equal |x| as in Definition 13.5.1. 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

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 going through 0. Note that there is no change in thiswhen you replace R3 with Rp.

13.7 Exercises1. Verify all the properties 13.3-13.10.

2. Compute the following

(a) 2(

1 2 3 −2)+6(

2 1 −2 7)

(b) −2(

1 2 −2)+6(

2 1 −2)

3. Find symmetric equations for the line through the points (2,2,4) and (−2,3,1) .Dumb idea but do it anyway.

4. Find symmetric equations for the line through the points (1,2,4) and (−2,1,1) .Dumb idea but do it anyway.

5. Symmetric equations for a line are given. Find parametric equations of the line. Thisgoes the right direction.

(a) x+13 = 2y+3

2 = z+7

(b) 2x−13 = 2y+3

6 = z−7

6. The first point given is a point contained in the line. The second point given is adirection vector for the line. Find parametric equations for the line, determined bythis information.

(a) (1,2,1) ,(2,0,3)

(b) (1,0,1) ,(1,1,3)

13.6. GEOMETRIC MEANING OF SCALAR MULTIPLICATION IN R? 28113.6 Geometric Meaning of Scalar Multiplication in R*As discussed earlier, 2 = (x),%2,x3) determines a vector. You draw the line from 0 toz placing the point of the vector on a. What is the length of this vector? The length ofthis vector is defined to equal |a| as in Definition 13.5.1. Thus the length of a equals4/ Xt +.x5 +23. When you multiply « by a scalar a, you get (Ox), 0x2, 0x3) and the lengthof this vector is defined as(iets (cxx2)° + (x3)”) = Joel y/ap +3 +23.Thus the following holds.|x| = |o| |ar].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 @ is negative, itcauses the resulting vector to point in the opposite direction while if @& > 0 it preserves thedirection the vector points. One way to see this is to first observe that if @ ~ 1, then x andaa are both points on the same line going through 0. Note that there is no change in thiswhen you replace R? with R?.13.7. Exercisesa. Verify all the properties 13.3-13.10.N. Compute the following(a) 2(1 2 3 -2)+6(2 1 -2 7)(b) -2( 1 2 -—2)+6(2 1 -2)3. Find symmetric equations for the line through the points (2,2,4) and (—2,3,1).Dumb idea but do it anyway.4. Find symmetric equations for the line through the points (1,2,4) and (—2,1,1).Dumb idea but do it anyway.5. Symmetric equations for a line are given. Find parametric equations of the line. Thisgoes the right direction.x+1 _ 2yt+3 __(a) “aS Set7(b) A= Bac -76. The first point given is a point contained in the line. The second point given is adirection vector for the line. Find parametric equations for the line, determined bythis information.(a) (1,2,1),(2,0,3)(b) (1,0,1),(1,1,3)(c) (1,2,0),(1, 1,0)