You do the work!

This entire class was an in-class activity. In it, you evaluated some expressions involving vector operations. More importantly, you proved the properties of vector operations summarized in the following theorem:

The following properties of the basic vector operations hold in any vector space $R^n$:
  1. vector addition is commutative and associative
  2. dot product is commutative, but dot product is *not* associative!
  3. $a\cdot\boldsymbol{u} + a\cdot\boldsymbol{v} = a\cdot(\boldsymbol{u} + \boldsymbol{v})$ where $a$ is scalar and $\boldsymbol{u}$ and $\boldsymbol{v}$ are vectors
  4. $(a\cdot b)\cdot \boldsymbol{u} = a\cdot (b\cdot \boldsymbol{u})$ where $a$ and $b$ are scalars and $\boldsymbol{u}$ is a vector
  5. $a\cdot (\boldsymbol{u}\cdot \boldsymbol{v}) = (a\cdot \boldsymbol{u})\cdot \boldsymbol{v} = \boldsymbol{u}\cdot (a\cdot \boldsymbol{v})$ where $a$ is scalar and $\boldsymbol{u}$ and $\boldsymbol{v}$ are vectors
  6. $(\boldsymbol{u} + \boldsymbol{v})\cdot \boldsymbol{w} = \boldsymbol{u}\cdot \boldsymbol{w} + \boldsymbol{v}\cdot \boldsymbol{w}$, where $\boldsymbol{u}$, $\boldsymbol{v}$ and $\boldsymbol{w}$ are vectors

You proved these in class! However, to make sure you remember how to do these kinds of proofs, I'll include here a proof that vector addition is commutative.

(point i: that vector addition is commutative) $$ \begin{array}{rcll} \boldsymbol{u} \cdot \boldsymbol{v} & = & [u_1\ u_2\ \cdots \ u_n] + [v_1\ v_2\ \cdots \ v_n]\\ &=& [u_1+v_1\ \ u_2+v_2\ \ \cdots \ \ u_n+v_n] & \text{ $\leftarrow$ definition of vector addition}\\ &=& [v_1+u_1\ \ v_2+u_2\ \ \cdots \ \ v_n+u_n] & \text{ $\leftarrow$ components are scalars, and addition of scalars is commutative}\\ & = & [v_1\ v_2\ \cdots \ v_n] + [u_1\ u_2\ \cdots \ u_n]& \text{ $\leftarrow$ definition of vector addition}\\ & = & \boldsymbol{v} \cdot \boldsymbol{u} \end{array} $$

Here's a proof of (vi) courtesy of Mr. Paris.

(point vi: vector addition and dot product distribute)

Christopher W Brown