Here is a question that has me stumped:
Use the geometric definition to find: $2 {\bf i} × ({\bf i}+{\bf j})$
Student solution manual says: By the definition of cross product, $2 {\bf i} × ({\bf i}+{\bf j})$ is in the direction of ${\bf k}$. The magnitude of it equals to the area of the parallelogram which is: $$ ||2{\bf i}||·||{\bf i}+{\bf j}||\sin(\pi/4) = 2\sqrt2·(\sqrt2/2)=2 $$ So $2{\bf i} × ({\bf i}+{\bf j}) =2{\bf k}$
I understand to use the geometric definition which is the formula $||v||·||w||\sin(\theta)·n$ where $\sin(\theta)$ is the angle between the vectors $v$ and $w$. The problem is why the $({\bf i}+{\bf j})$ is a rotation of $45$ degrees between the $x$ and $y$ axis.
Using the linearity of the cross product and the cross products of the canonical basis vectors, one can simply compute $$ 2{\bf i}×({\bf i}+{\bf j})=2{\bf i}×{\bf i}+2{\bf i}×{\bf j}=0+2{\bf k} $$
For any two vectors $v$, $w$ of equal length, $v+w$, or in general $\|w\|·v+\|v\|·w$, -- if not zero -- is a vector in the direction of the angle bisector between $v$ and $w$.
Since the angle between the unit vectors ${\bf i}$ and ${\bf j}$ is $90°$, the bisected angle is $45°$ between ${\bf i}$ and ${\bf i}+{\bf j}$ resp. ${\bf i}+{\bf j}$ and ${\bf j}$.