The Magnitude and Direction of a Vector

The magnitude (length) of the vector $\vec{a} = (a_1, a_2, a_3)$ in space is defined as:

where $a_1, a_2, a_3$ are the components of the vector $\vec{a}$ on the three coordinate axes.

The vector in the same direction as the nonzero vector $\vec{a}$ with magnitude 1 is called the unit vector of $\vec{a}$, written $\vec{e}_a$:

where $|\vec{a}|$ is the magnitude of the vector $\vec{a}$.

The cosines of the angles $\alpha, \beta, \gamma$ that the nonzero vector $\vec{a} = (a_1, a_2, a_3)$ makes with the positive directions of the three coordinate axes are called the direction cosines of that vector:

The sum of the squares of the direction cosines equals 1:

where $\alpha, \beta, \gamma$ are the angles that the vector $\vec{a}$ makes with the positive directions of the $x, y, z$ axes respectively.