Consider a vector as shown below on the x-y-z plane. The angles made by this line with the +ve direactions of the coordinate axes: θ_{x}, θ_{y}, and θ_{z} are used to find the direction cosines of the line: cos θ_{x}, cos θ_{y}, and cos θ_{z}. Likewise, the direction cosine of θ_{x}, θ_{y}, and θ_{z} in the opposite direction of the line are –cos θ_{x}, -cos θ_{y}, and -cos θ_{z}

Note that the angles θ_{x}, θ_{y}, and θ_{z} are the direction angles that satisfy the condition: 0<θ_{x}, θ_{y},θ_{z}<π. Also, the sum of direction angles: θ_{x} + θ_{y} + θ_{z}≠ 2π because these angles do not lie on the same plane.

For a line on the x-axis, the direction cosines are cos 0, cos π/2, cos π/2 = 1, 0, 0.

For a line on the y-axis, the direction cosines are cos π/2, cos 0, cos π/2 = 0, 1, 0.

For a line on the x-axis, the direction cosines are cos π/2, cos π/2,cos 0 =0, 0, 1.

**Consider a point P (x, y, z) on a 3D plane such that OP = $\overrightarrow{r}$ and let l, m and n be the direction cosine of $\overrightarrow{r}$ then we can conclude that: x=l$\overrightarrow{r}$, y=m$\overrightarrow{r}$, and z=n$\overrightarrow{r}$.**

Hence,

$l=\frac{x}{\overrightarrow{r}}=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}$

$m=\frac{y}{\overrightarrow{r}}=\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}$

$n=\frac{z}{\overrightarrow{r}}=\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}$

**The equation of the line can be written as: $l^{2}+m^{2}+n^{2}=1$**

Proof:

We know that l = cosθ$_{x}$, m = cosθ$_{y}$, and n = cosθ$_{z}$

And, cos$^{2}$θ$_{x}$ + cos$^{2}$θ$_{y}$ + cos$^{2}$θ$_{z}$ = 1

cos$^{2}$θ$_{x}$ + cos$^{2}$θ$_{y}$ + cos$^{2}$θ$_{z}$= $(\frac{x}{\overrightarrow{r}})^{2}+(\frac{y}{\overrightarrow{r}})^{2}+(\frac{z}{\overrightarrow{r}})^{2}=\frac{x^{2}+y^{2}+y^{2}}{r^{2}}=\frac{r^{2}}{r^{2}}=1$

Now consider two points in a line segment $\overrightarrow{AB}$, with the coordinates A (x$_{1}$, y$_{1}$, z$_{1}$) and B (x$_{2}$, y$_{2}$, z$_{2}$). Then, their direction cosines can be written a$(\frac{x_{2}-x_{1}}{\overrightarrow{r}}, \frac{y_{2}-y_{1}}{\overrightarrow{r}}, \frac{z_{2}-z_{1}}{\overrightarrow{r}})$.

In this case, $\overrightarrow{\vert r\vert}=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

**Example: A line makes 30°, 60° and 90° with the x, y and z axes respectively. Find their direction cosines.**

Solution:

l = cosθ$_{x}$ = cos 30° = $\frac{\sqrt{3}}{2}$

m = cosθ$_{y}$ = cos 60° = $\frac{1}{2}$

and n = cosθ$_{z}$= cos 90° = 0

The direction cosines are: $(\frac{\sqrt{3}}{2}, \frac{1}{2}, 0)$

**Example: Consider a point P $(\sqrt{3},1, 2\sqrt{3} )$ in a 3D space, find the direction cosines of $\overrightarrow{OP}$**

Solution: The direction ratios of point P= $(\sqrt{3},1, 2\sqrt{3} )$

Recall that $l=\frac{x}{\overrightarrow{r}}=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}$

$m=\frac{y}{\overrightarrow{r}}=\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}$

$n=\frac{z}{\overrightarrow{r}}=\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}$

In this case, $\overrightarrow{r}=\overrightarrow{OP}$ and (x, y, z) = $(\sqrt{3},1, 2\sqrt{3} )$

$\vert \overrightarrow{OP}\vert =\sqrt{(\sqrt{3}-0)^{2}+(1-0)^{2}+(2\sqrt{3}-0)^{2}}=\sqrt{3+1+12}=4$

The direction cosine of $\overrightarrow{OP}$ are: $(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{2\sqrt{3}}{4})=(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2}

)$