What are the vectors vec a and vec b if 1^° vec a×vec b=vec…
What are the vectors $\vec a$ and $\vec b$ if $1^\circ\ \ \vec a\times\vec b=\vec 0,\qquad\qquad 2^\circ\ \ \vec a\times\vec b=\vec b\times\vec a\,?$
Explanation of Method
When is the vector product zero, and when does it commute?
Step 1: $1^\circ$ $\vec a\times\vec b=\vec 0$
\[ |\vec a\times\vec b|=|\vec a||\vec b|\sin\varphi=0\iff \vec a=\vec 0\ \text{ or }\ \vec b=\vec 0\ \text{ or }\ \sin\varphi=0. \]
Here $\sin\varphi=0$ means $\varphi=0^\circ$ or $180^\circ$: the vectors are collinear. Geometrically, the parallelogram on $\vec a,\vec b$ has zero area.
\[\vec a\times\vec b=\vec 0\iff\vec a\parallel\vec b\quad(\text{including }\vec a=\vec 0\text{ or }\vec b=\vec 0)\]
Step 2: $2^\circ$ $\vec a\times\vec b=\vec b\times\vec a$
By anti-commutativity $\vec b\times\vec a=-\vec a\times\vec b$, so
\[ \vec a\times\vec b=-\vec a\times\vec b\ \Longrightarrow\ 2\,\vec a\times\vec b=\vec 0\ \Longrightarrow\ \vec a\times\vec b=\vec 0. \]
\[\vec a\times\vec b=\vec b\times\vec a\iff\vec a\parallel\vec b\quad(\text{same answer as }1^\circ)\]
Result
\[\boxed{\,\vec a\times\vec b=\vec 0\iff\vec a\parallel\vec b\quad(\text{including }\vec a=\vec 0\text{ or }\vec b=\vec 0)\,}\]
\[\boxed{\,\vec a\times\vec b=\vec b\times\vec a\iff\vec a\parallel\vec b\quad(\text{same answer as }1^\circ)\,}\]