Chapter 10: Q .9. (page 811)

9. Let $u$ be a nonzero vector.
(a) Show that $u \cdot v = u \cdot w$ does not necessarily imply that $v = w$ .
(b) What geometric relationship must $u$ , $v$ , and $w$ satisfy if $u \cdot v = u \cdot w$ ?

Short Answer

Expert verified

Part a)Proved

Part b)

Step by step solution

Part a):Given information

$u . v = u . w$ (GIven)

Step 2:Explaination Part b)

$Consider the non-zero vector u .$

$Assume that u = i, v = i and w = i + j .$

$The vectors v = i and w = i + jare not equal.$

$The dot product u \cdot v is:$

$u \cdot v = i \cdot i$

$The dot product u \cdot w is:$

$u \cdot w = i \cdot (i + j)$

$= i \cdot i + i \cdot j$

$= 1 + 0$

$= 1$

$Therefore, for the vectors u = i, v = i and w = i + j; u \cdot v = u \cdot w does not necessarily imply that$

$v = w$

Step 3:Given information Part b)

given $u . v = u . w$

Step 2:Explaiination Part b)

$The objective is to determine the geometric relationship the vectors u, v and w satisfy if$

$u \cdot v = u \cdot w$

$The condition that the vectors must satisfy for u \cdot v = u \cdot w is:$

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

$u \cdot v - u \cdot w = 0 (Transposing)$

$u \cdot (v - w) = 0 (Dot product is distributive)$

$The dot product of vectors u and v - w is zero.$

$The condition u \cdot (v - w) = 0 gives that the vectors u and v - w should be orthogonal to hold$

$u \cdot v = u \cdot w$

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9. Let $u$ be a nonzero vector.
(a) Show that $u \cdot v = u \cdot w$ does not necessarily imply that $v = w$ .
(b) What geometric relationship must $u$ , $v$ , and $w$ satisfy if $u \cdot v = u \cdot w$ ?

Short Answer

Step by step solution

Part a):Given information

Step 2:Explaination Part b)

Step 3:Given information Part b)

Step 2:Explaiination Part b)

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Most popular questions from this chapter

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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

Step by step solution

Part a):Given information

Step 2:Explaination Part b)

Step 3:Given information Part b)

Step 2:Explaiination Part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Pure Maths

Probability and Statistics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

9. Let $u$ be a nonzero vector.
(a) Show that $u \cdot v = u \cdot w$ does not necessarily imply that $v = w$ .
(b) What geometric relationship must $u$ , $v$ , and $w$ satisfy if $u \cdot v = u \cdot w$ ?