Multiplication of Vector by a Scalar

Multiplying a vector $\vec A$ by an ordinary number (a scalar) $\lambda$ produces another vector, $\lambda\vec A$, whose magnitude is $|\lambda|$ times the magnitude of $\vec A$. The direction depends on the sign of $\lambda$: if $\lambda$ is positive, $\lambda\vec A$ points the same way as $\vec A$; if $\lambda$ is negative, $\lambda\vec A$ points exactly opposite to $\vec A$.

This simple rule is exactly what defines several central physical quantities as 'a scalar times a vector':

• Force: $\vec F = m\vec a$. Since mass $m$ is always a positive scalar, the force is always in the same direction as the acceleration $\vec a$.
• Linear momentum: $\vec p = m\vec v$. Again $m>0$, so momentum always points along the velocity $\vec v$. …