Scalar Multiplication of a Vector

Definition. For a vector $\vec a$ and a scalar (real number) $m$, the vector $m\vec a$ is called the scalar multiple of $\vec a$ by $m$.

Effect on magnitude and direction.

• If $m=0$, then $m\vec a=\vec 0$ (the zero vector) regardless of $\vec a$.
• If $m>0$, then $m\vec a$ points in the same direction as $\vec a$; if $m<0$, it points in the opposite direction. So $\vec a$ and $m\vec a$ are like vectors when $m>0$ and unlike vectors when $m<0$.
• The magnitude scales exactly as expected: $|m\vec a|=|m|\,|\vec a|.$ …