even_and_odd_and_vector_quiz.tex

\yourname

\activitytitle{Quiz on even and odd and three--dimensional vectors}{20 points}

\noindent Work hard to write really nice proofs.

\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 $n = 2k+1$.}

\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. Scalar product for 3--dimensional vectors}{Let $c$ be a real number and let $\vec{a} = \tvec{a}$ be a 3--dimensional vector.  The {\em scalar product} of $c$ and $\vec{a}$ is defined as:
\[
    c\vec{a} = \vect{ca_1,ca_2,ca_3}.
\]}

\showNN{Show that the product of two odd numbers is odd.}{5in}

\showNN{On the back of this sheet, show that the scalar product is distributive over vector addition. That is, show that $c(\vec{a} \oplus \vec{b}) = c\vec{a} \oplus c\vec{b}$.}{0in}


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