even_and_odd_and_vector_quiz_3.tex

\yourname

\activitytitle{Quiz on even and odd and three--dimensional vectors}{}

\noindent Write really nice proofs.
Use good form.  Start in the right place, take small steps, justify each step, and generalize appropriately.

\definitionNN{1. Even}{An integer $n$ is {\em even} if there exists an integer $k$ for which $n = 2k$.}

\definitionNN{2. Odd}{An integer $n$ is {\em odd} if there exists an integer $k$ for which $n = 2k+1$.}

\showNN{Show that the difference of two odd numbers is even.}{0in}

\pagebreak

\definitionNN{3. Three--dimensional vector}{A three--dimensional vector is an ordered triple $\vect{ a_1, a_2, a_3 }$, where $a_1, a_2,$ and $a_3$ are real numbers.}

\definitionNN{4. Sum of 3--dimensional vectors}{The sum of 3--dimensional vectors $\vect{ a_1, a_2, a_3 }$ and $\vect{ b_1, b_2, b_3 }$ is the 3--dimensional vector $\vect{ a_1+b_1, a_2+b_2, a_3+b_3 }$.  We write $\vec{a} \oplus \vec{b}$, using a new symbol so we don't confuse addition of vectors with addition of real numbers.}

\definitionNN{5. 1234 product of 3--dimensional vectors}{The 1234 product of 3--dimensional vectors $\vect{ a_1, a_2, a_3 }$ and $\vect{ b_1, b_2, b_3 }$ is the real number $1 a_1 b_1 + 2 a_2 b_2 + 3 a_3 b_3 + 4 a_1 b_3$. 
The 1234 product of vectors $\vec{a}$ and $\vec{b}$ is denoted $\vec{a} \star \vec{b}$.}


\showNN{Show that the 1234 product is distributive over vector addition. 
That is, show that $\vec{a} \star (\vec{b} \oplus \vec{c}) = \vec{a} \star \vec{b} + \vec{a} \star \vec{c}$ for all vectors $\vec{a}, \vec{b},$ and $\vec{c}$.}{0in}


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