Question

9. Let $\mathbf{u}$be a nonzero vector.

(a) Show that $\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathbf{w}$does not necessarily imply that $\mathbf{v}=\mathbf{w}$.

(b) What geometric relationship must $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$satisfy if $\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathbf{w}$?

Short answer

Part a)Proved

Part b)

Step-by-step solution

Part a):Given information

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

Step 2:Explaination Part b)

$\text{Consider the non-zero vector}\mathbf{u}\text{.}$

$\text{Assume that}\mathbf{u}=\mathbf{i},\mathbf{v}=\mathbf{i}\text{and}\mathbf{w}=\mathbf{i}+\mathbf{j}\text{.}$

$\text{The vectors}\mathbf{v}=\mathbf{i}\text{and}\mathbf{w}=\mathbf{i}+\mathbf{jare}\text{not equal.}$

$\text{The dot product}\mathbf{u}·\mathbf{v}\text{is:}$

$\mathbf{u}·\mathbf{v}=\mathbf{i}·\mathbf{i}$

$\text{The dot product}\mathbf{u}·\mathbf{w}\text{is:}$

$\mathbf{u}·\mathbf{w}=\mathbf{i}·(\mathbf{i}+\mathbf{j})$

$=\mathbf{i}·\mathbf{i}+\mathbf{i}·\mathbf{j}$

$=1+0$

$=1$

$\text{Therefore, for the vectors}\mathbf{u}=\mathbf{i},\mathbf{v}=\mathbf{i}\text{and}\mathbf{w}=\mathbf{i}+\mathbf{j};\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathbf{w}\text{does not necessarily imply that}$

$\mathit{v}\mathbf{=}\mathit{w}$

Step 3:Given information Part b)

given$u.v=u.w$

Step 2:Explaiination Part b)

$\text{The objective is to determine the geometric relationship the vectors}\mathbf{u},\mathbf{v}\text{and}\mathbf{w}\text{satisfy if}$

$\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathbf{w}$

$\text{The condition that the vectors must satisfy for}\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathbf{w}\text{is:}$

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

$\mathbf{u}·\mathbf{v}-\mathbf{u}·\mathbf{w}=0\text{(Transposing)}$

$\mathbf{u}·(\mathbf{v}-\mathbf{w})=0\text{(Dot product is distributive)}$

$\text{The dot product of vectors}\mathbf{u}\text{and}\mathbf{v}-\mathbf{w}\text{is zero.}$

$\text{The condition}\mathbf{u}·(\mathbf{v}-\mathbf{w})=0\text{gives that the vectors}\mathbf{u}\text{and}\mathbf{v}-\mathbf{w}\text{should be orthogonal to hold}$

$\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathit{w}$