Some Properties and Results

For any vectors $\vec a,\vec b$ and scalars $m,n$, scalar multiplication obeys the same rules as ordinary numbers:

1. $m(n\vec a)=(mn)\vec a=n(m\vec a)$ — associativity of scalar multiples.
2. $(m+n)\vec a=m\vec a+n\vec a$ — distributes over scalar addition.
3. $m(\vec a+\vec b)=m\vec a+m\vec b$ — distributes over vector addition. Vector addition itself obeys three further laws, all provable from the triangle/parallelogram law:

• Associativity: for any three vectors $\vec a,\vec b,\vec c$, $(\vec a+\vec b)+\vec c=\vec a+(\vec b+\vec c)$.
• Identity: for any vector $\vec a$, $\vec a+\vec 0=\vec 0+\vec a=\vec a$.
• Inverse: for any vector $\vec a$, $\vec a+(-\vec a)=(-\vec a)+\vec a=\vec 0$ — every vector has an additive inverse.
• Commutativity: $\vec a+\vec b=\vec b+\vec a$. Proof sketch: complete the parallelogram $OACB$ with $\vec{OA}=\vec a,\vec{OB}=\vec b$ as adjacent sides. Going via $A$: $\vec a+\vec b=\vec{OA}+\vec{AC}=\vec{OC}$. Going via $B$: $\vec b+\vec a=\vec{OB}+\vec{BC}=\vec{OC}$ (using $\vec{OA}=\vec{BC}$, opposite sides of the parallelogram). Both routes reach the same diagonal $\vec{OC}$, so $\vec a+\vec b=\vec b+\vec a$. …