Multiplication of a Vector by a Real Number

A vector $\vec A$ can be multiplied by an ordinary real number $\lambda$ (called a scalar in

this context) to produce a new vector $\lambda\vec A$. The rules governing this

scalar multiplication are:

• The magnitude of $\lambda \vec A$ is $|\lambda|$ times the magnitude of $\vec A$, i.e. $|\lambda \vec A| = |\lambda|\,A$.
• If $\lambda > 0$, the vector $\lambda\vec A$ points in the same direction as $\vec A$.
• If $\lambda < 0$, the vector $\lambda\vec A$ points in the direction exactly opposite to $\vec A$.

So multiplying by a positive number greater than 1 stretches the vector while keeping its

direction unchanged; multiplying by a positive number between 0 and 1 shrinks it, again

keeping the direction unchanged; and multiplying by a negative number both scales the

magnitude and reverses the direction. The special case $\lambda = -1$ simply reverses the

vector without changing its length, giving the negative of a vector, $-\vec A$ — this is

exactly what makes vector subtraction possible (Section 3.5): $\vec A - \vec B$ is defined as

$\vec A + (-\vec B)$.

Another important special case is $\lambda = 1/A$ (where $A = |\vec A|$ and $A \ne 0$): this

produces a vector of magnitude exactly 1, pointing in the same direction as $\vec A$, called

the unit vector along $\vec A$ and written $\hat A$: …