Q .9. - Calculus

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

Step-by-Step Solution

Answer

Part a)Proved

Part b)

Step 1 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" style="max-width: none; vertical-align: -4px;" $\mathbf{u}·\mathbf{v}=\mathbf{u}·\mathit{w}$

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

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

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

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

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