# Check properties of dot product problem

• May 5th 2013, 12:53 PM
emccormick
Check properties of dot product problem
Check please. Just need to make sure I am doing this right.

Given$\displaystyle \vec{u}\cdot\vec{u}=8,\;\vec{v}\cdot\vec{v}=6,\; \vec{u}\cdot\vec{v}=7$

Find: $\displaystyle (3u-v)\cdot (u-3v)$

$\displaystyle (3u-v)\cdot (u-3v)\implies \\(3u-v)\cdot u-[(3u-v)\cdot 3v]\implies \\3u\cdot u-v\cdot u-[3u\cdot 3v-v\cdot 3v]\implies \\3(u\cdot u)-v\cdot u-9(u\cdot v)+3(v\cdot v)\implies \\3(8)-7-9(7)+3(6)\implies \\(3u-v)\cdot (u-3v)=-28$
• May 5th 2013, 01:32 PM
Plato
Re: Check properties of dot product problem
Quote:

Originally Posted by emccormick
Check please. Just need to make sure I am doing this right.
Given$\displaystyle \vec{u}\cdot\vec{u}=8,\;\vec{v}\cdot\vec{v}=6,\; \vec{u}\cdot\vec{v}=7$
Find: $\displaystyle (3u-v)\cdot (u-3v)$
$\displaystyle (3u-v)\cdot (u-3v)\implies \\(3u-v)\cdot u-[(3u-v)\cdot 3v]\implies \\3u\cdot u-v\cdot u-[3u\cdot 3v-v\cdot 3v]\implies \\3(u\cdot u)-v\cdot u-9(u\cdot v)+3(v\cdot v)\implies \\3(8)-7-9(7)+3(6)\implies \\(3u-v)\cdot (u-3v)=-28$

Yes that is correct. But note that it is easier to see

$\displaystyle (3\vec{u}-\vec{v})\cdot(\vec{u}-3\vec{u})=3\vec{u}\cdot\vec{u}-10\vec{u}\cdot\vec{v}+3\vec{v}\cdot\vec{v}$
• May 5th 2013, 01:41 PM
emccormick
Re: Check properties of dot product problem
Thanks. Thought it was, but no solutions to the even ones. Whee.