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9. Let ube a nonzero vector.

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

(b) What geometric relationship must u, v, and wsatisfy if u·v=u·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·vis:

u·v=i·i

The dot productu·wis:

u·w=i·(i+j)

=i·i+i·j

=1+0

=1

Therefore, for the vectorsu=i,v=iandw=i+j;u·v=u·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·v=u·w

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

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

u·v-u·w=0(Transposing)

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

The dot product of vectorsuandv-wis zero.

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

u·v=u·w

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