4.3 Direction of a Vector

For any non-zero vector \(A\), we obtain a unit vector called the direction of \(A\) by dividing \(A\) by its own length, i.e \[\text {Direction of }\hspace {0.3cm} A = \frac {A}{ \left |A\right | }\]

Direction Cosines

Let \(\alpha \) be the angle between the vector \(\overrightarrow {OP}\) and the positive \(x -\) axis, \(\beta \) the angle between the \(y - \) axis and \(\overrightarrow {OP}\) and \(\gamma \) be the angle between the \(z - \) axis and \(\overrightarrow {OP}\). The angles \(\alpha , \beta , \gamma \) are called the direction angle of the vector \(\overrightarrow {OP}\).

\[\cos \alpha = \frac {x_0}{\sqrt {x_0^2 + y_0^2 + z_0^2}}\]

\[\cos \beta = \frac {y_0}{\sqrt {x_0^2 + y_0^2 + z_0^2}}\]

\[\cos \gamma = \frac {z_0}{\sqrt {x_0^2 + y_0^2 + z_0^2}}\]

The numbers \(\cos \alpha , \cos \beta \) and \(\cos \gamma \) are called the direction cosines of \(\overrightarrow {OP}\).

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