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Chapter 10: Q.61E (page 802)

61.

Let cand dbe a scalars and let vbe a vector in 鈩�3. Show that the following distributive property holds:

(c+d)v=cv+dv.

Short Answer

Expert verified

What do we mean when we say S is linearly independent., S is closed in both addition and scalar multiplication.

Step by step solution

01

Introduction.

Consider the vector vin R3and cand dare any two scalars.

The objective is to prove that (c+d)v=cv+dv.

If there are two vectors u=x1,y1,z1

and cis any scalar, then given bycu=cx1,cy1,cz1
02

Given Information.

Now, let v=v1,v2,v3鈥�鈥�(1)

Therefore,

Now,


(c+d)v=(c+d)鉄�v1,v2,v3鉄�=鉄�(c+d)v1,(c+d)v2,(c+d)v3鉄�=鉄�cv1+dv1,cv2+dv2,cv3+dv3鉄�

Therefore,(c+d)v=鉄�cv1+dv1,cv2+dv2,cv3+dv3鉄�

03

Explanation (part a).

Now, rewrite (c+d)v=cv1+dv1,cv2+dv2,cv3+dv3


(c+d)v=鉄�cv1,cv2,cv3鉄�+鉄�dv1,dv2,dv3鉄�=c鉄�v1,v2,v3鉄�+c鉄�v1,v2,v3鉄�

Hence,(c+d)v=cv1,v2,v3+cv1,v2,v3..........(2)

04

Explanation (part b).

Using (1)in(2)

Therefore,

(c+d)v=cv+dv

Hence, it is proved that(c+d)v=cv+dv

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Most popular questions from this chapter

If u, v and w are three vectors in 鈩�3, what is wrong with the expression u脳v脳w?

Find u+vandu-valso sketch u,v,u+v,u-v

u=4,0andv=0,3

In Exercises 36鈥�41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(4,鈭�2),Q(鈭�2,0),R(1,鈭�5)

(Hint: Think of the xy-plane as part of 鈩�3.)

In Exercises 24-27, find compuv, projuv, and the component of v orthogonal tou.

u=1,2,v=3,5

In Exercises 24-27, find compuv,projuv and the component of v orthogonal tou.

role="math" localid="1649693816584" u=3,1,-2,v=-6,-2,4
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