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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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Most popular questions from this chapter

In Exercises 36–41 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(1,6),Q(0,3),R(5,4)

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

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Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is k=1.

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(f) True or False: k=0nak=a0an+k=1n1ak.

(g) True or False: k=110ak2=k=110ak2.

(h) True or False: k=1nex2=exex+12ex+16.

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