Direction cosines, direction ratios, and lines

A line in three-dimensional space has a direction, and any two parallel lines share that direction. We capture direction quantitatively by the angles the line makes with the three coordinate axes , the direction cosines , or by any triple proportional to them , the direction ratios.

Direction cosines

If a directed line makes angles $\alpha, \beta, \gamma$ with the positive $x$-, $y$-, $z$-axes, its direction cosines are $l = \cos\alpha, \quad m = \cos\beta, \quad n = \cos\gamma.$

These satisfy $l^2 + m^2 + n^2 = 1.$

For an undirected line, the direction cosines are defined up to a simultaneous sign change , $(l, m, n)$ and $(-l, -m, -n)$ describe the same line, just two opposite orientations.

Direction ratios

Any triple $(a, b, c)$ proportional to $(l, m, n)$ , i.e. $a = kl$, $b = km$, $c = kn$ for some $k \ne 0$ , is called a set of direction ratios of the line. Direction ratios are not unique; you can multiply by any non-zero scalar.

Given direction ratios $(a, b, c)$, direction cosines are recovered by $l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}.$

From two points

A line through points $P = (x_1, y_1, z_1)$ and $Q = (x_2, y_2, z_2)$ has direction $\vec{PQ} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)$. So direction ratios are simply $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$ , coordinate differences.

Why $l^2 + m^2 + n^2 = 1$

If $\vec u$ is the unit vector along the line, $\vec u = (l, m, n)$. Then $|\vec u|^2 = l^2 + m^2 + n^2 = 1$. This identity replaces a key piece of two-dimensional intuition: in 2D the angle of a line was a single number $\theta$ with $\cos^2\theta + \sin^2\theta = 1$; in 3D you need three direction cosines satisfying one constraint.

Conditions for special lines

• Line parallel to $x$-axis: $(l, m, n) = (\pm 1, 0, 0)$.
• Line in the $xy$-plane: $n = 0$, so $l^2 + m^2 = 1$.
• Line through origin and $(a, b, c)$: direction ratios $(a, b, c)$.

Two lines: angle and special positions

If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the angle between them is given by $\cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}.$ (The absolute value forces $\theta \in [0, \pi/2]$.)

Perpendicular: $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.

Parallel: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$.

Worked examples

Example 1. Find the direction cosines of the line joining $(1, 2, 3)$ and $(4, 6, 3)$.

Direction ratios $(3, 4, 0)$. Magnitude $\sqrt{9 + 16 + 0} = 5$. Direction cosines $(3/5, 4/5, 0)$.

Example 2. Show that the line making equal angles with all three positive coordinate axes has direction cosines $(1/\sqrt 3, 1/\sqrt 3, 1/\sqrt 3)$.

$l = m = n$, so $3 l^2 = 1$, $l = \pm 1/\sqrt 3$.

Example 3. Find the angle between the lines with direction ratios $(1, 2, 1)$ and $(2, 3, 1)$.

$\cos\theta = \dfrac{|2 + 6 + 1|}{\sqrt 6 \cdot \sqrt{14}} = \dfrac{9}{\sqrt{84}} = \dfrac{9}{2\sqrt{21}}$.

Example 4. Find $\lambda$ if the lines with direction ratios $(1, \lambda, 3)$ and $(2, 1, -1)$ are perpendicular.

$2 + \lambda - 3 = 0$, so $\lambda = 1$.

Example 5. A line makes angles $60^\circ, 60^\circ, \gamma$ with the axes. Find $\gamma$.

$\cos^2 60^\circ + \cos^2 60^\circ + \cos^2\gamma = 1 \Rightarrow 1/4 + 1/4 + \cos^2\gamma = 1$, so $\cos^2\gamma = 1/2$, $\gamma = 45^\circ$ or $135^\circ$.

Example 6. Are the lines through $(1, 1, 1) \to (4, 4, 4)$ and through $(2, 0, 0) \to (8, 6, 6)$ parallel?

Direction ratios: $(3, 3, 3) \propto (1, 1, 1)$; $(6, 6, 6) \propto (1, 1, 1)$. Same direction. Yes, parallel.

Try it yourself

1. Direction cosines of a line with direction ratios $(2, 3, 6)$.
2. Find the angle between lines with direction ratios $(1, 1, 2)$ and $(2, 1, 1)$.
3. Show that lines with direction ratios $(2, 3, -1)$ and $(1, -2, -4)$ are perpendicular.
4. Find $\lambda$ if lines with direction ratios $(3, 2, \lambda)$ and $(1, 1, -2)$ are perpendicular.
5. A line makes angles $45^\circ$ and $60^\circ$ with the $x$- and $y$-axes. Find the angle with the $z$-axis.
6. Show that the line joining $(0, 0, 0)$ and $(1, 1, 1)$ makes equal angles with all coordinate axes.
7. Find direction cosines for the $z$-axis.
8. Find direction ratios for the line joining $(2, -3, 1)$ to $(5, 1, -2)$.
9. Two lines have direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$. Show $\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2$.
10. Are the lines through $(1, 0, 0), (2, 1, 0)$ and through $(0, 0, 0), (1, 1, 0)$ parallel?
11. Find direction ratios for the line that is equally inclined to the coordinate axes.
12. A line lies in the $xy$-plane and makes a $30^\circ$ angle with the $x$-axis. Find its direction cosines.
13. Find $\mu$ if lines with direction ratios $(1, 2, 3)$ and $(-1, \mu, 1)$ are parallel.
14. Find the angle between the body diagonal of a cube and one of its edges.

Pitfalls and tricks

• $l^2 + m^2 + n^2 = 1$ for direction cosines, not for direction ratios.
• Direction ratios are not unique , they scale.
• For perpendicularity, the dot product of direction ratios is zero.
• For parallelism, direction ratios are proportional (same up to scaling).
• Angle between lines is in $[0, \pi/2]$ by convention. Always take absolute value of $\cos$.