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Chapter 10: Q .9. (page 811)

9. Let ube a nonzero vector.

(a) Show that u路v=u路wdoes not necessarily imply that v=w.

(b) What geometric relationship must u, v, and wsatisfy if u路v=u路w?

Short Answer

Expert verified

Part a)Proved

Part b)

Step by step solution

01

Part a):Given information

u.v=u.w(GIven)

02

Step 2:Explaination Part b)

Consider the non-zero vectoru.

Assume thatu=i,v=iandw=i+j.

The vectorsv=iandw=i+jarenot equal.

The dot productu路vis:

u路v=i路i

The dot productu路wis:

u路w=i路(i+j)

=i路i+i路j

=1+0

=1

Therefore, for the vectorsu=i,v=iandw=i+j;u路v=u路wdoes not necessarily imply that

v=w

03

Step 3:Given information Part b)

givenu.v=u.w

04

Step 2:Explaiination Part b)

The objective is to determine the geometric relationship the vectorsu,vandwsatisfy if

u路v=u路w

The condition that the vectors must satisfy foru路v=u路wis:

width="76" height="20" role="math">u路v=u路w

u路v-u路w=0(Transposing)

u路(v-w)=0(Dot product is distributive)

The dot product of vectorsuandv-wis zero.

The conditionu路(v-w)=0gives that the vectorsuandv-wshould be orthogonal to hold

u路v=u路w

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