Magnitude of a Vector

The magnitude (also called the size or length) of a vector $\overrightarrow{AB}$ is written

$|\overrightarrow{AB}|$ and is defined simply as the length of the segment $AB$, i.e.

$|\overrightarrow{AB}| = l(AB)$. If vectors are written as $\bar a,\bar b,\bar c$, their magnitudes are

written $|\bar a|,|\bar b|,|\bar c|$. Since length can never be negative, $|\bar a|\ge 0$ for every vector

$\bar a$, and the magnitude of a vector does not depend on which way it points -- $\bar a$ and $-\bar a$

always have the same magnitude.

Worked example. To find the magnitude of $\bar a = \hat i-2\hat j+4\hat k$, add the squares of the

components under a square root: $|\bar a|=\sqrt{1^2+(-2)^2+4^2}=\sqrt{1+4+16}=\sqrt{21}$. The same rule gives

$|\bar b|=\sqrt{4^2+(-3)^2+(-7)^2}=\sqrt{16+9+49}=\sqrt{74}$ for $\bar b=4\hat i-3\hat j-7\hat k$, and for a …