Mathematics · Ch 11 — Three-Dimensional Geometry

Direction Cosines and Direction Ratios of a Line

11.2

Direction Cosines and Direction Ratios of a Line

Direction Angles and Direction Cosines

When a directed line passes through the origin and makes angles $\alpha$ , $\beta$ , and $\gamma$ with the positive $x$ , $y$ , and $z$ -axes respectively, these angles are the direction angles of the line. Their cosines — $\cos\alpha$ , $\cos\beta$ , $\cos\gamma$ — are the direction cosines.

Reversing the direction of the line replaces each direction angle by its supplement ( $\pi - \alpha$ , $\pi - \beta$ , $\pi - \gamma$ ). Since $\cos(\pi - \theta) = -\cos\theta$ , all three direction cosines change sign.

Note

A line can be extended in two opposite directions, giving two sets of direction cosines that differ only in sign. To obtain a unique set, treat the line as a directed line.

For a directed line, the unique direction cosines are denoted $l$ , $m$ , $n$ , where:

l = \cos\alpha,\quad m = \cos\beta,\quad n = \cos\gamma

Important

If the line does not pass through the origin, draw a line through the origin parallel to it. Parallel lines have the same direction cosines.

Direction Ratios

Any three numbers proportional to the direction cosines of a line are its direction ratios. If $l$ , $m$ , $n$ are direction cosines and $a$ , $b$ , $c$ are direction ratios, there exists a non-zero real number $\lambda$ such that:

a = \lambda l,\quad b = \lambda m,\quad c = \lambda n

Tip

Direction ratios are not unique — any scalar multiple of a set of direction ratios is also a valid set.

Relation Between Direction Cosines and Direction Ratios

Since the two sets are proportional, write $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k$ , so:

l = ak,\quad m = bk,\quad n = ck \quad \ldots (1)

Using the fundamental identity $l^2 + m^2 + n^2 = 1$ and substituting (1):

k^2(a^2 + b^2 + c^2) = 1 \quad\Longrightarrow\quad k = \pm \frac{1}{\sqrt{a^2 + b^2 + c^2}}

l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}},\quad m = \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}},\quad n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}}

The sign is chosen depending on the desired orientation, and all three signs must be taken consistently.

Watch out

A common mistake is to take different signs for different direction cosines. Since $k$ is a single constant, the sign must be the same for $l$ , $m$ , and $n$ .

Properties of Direction Ratios

›Proof

Property 1: If $a$ , $b$ , $c$ are direction ratios of a line, then so are $ka$ , $kb$ , $kc$ for any non-zero $k$ .

With $a = \lambda l$ , $b = \lambda m$ , $c = \lambda n$ , we get $ka = k\lambda l$ , $kb = k\lambda m$ , $kc = k\lambda n$ . Since $k\lambda \neq 0$ , these are proportional to $l$ , $m$ , $n$ , hence direction ratios of the same line.

›Proof

Property 2: Any two sets of direction ratios of a line are proportional.

…

Figure 11.1A directed line OL from the origin into the first octant of the X, Y, Z axes, showing the direction angles alpha, beta and gamma it makes with the three axes and its projections x, y, z.

Fig. 11.1 — A directed line OL from the origin into the first octant of the X, Y, Z axes, showing the direction angles alpha, beta and gamma it makes with the three axes and its projections x, y, z.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 11.1 is the foundational picture for the entire chapter. It shows a single directed line segment $OL$ starting at the origin $O$ and running into the first octant — the region where all three coordinates are positive. The line is drawn as an arrow, with the arrowhead at $L$ , to make it a directed line. The point $P$ lies somewhere on $OL$ , and the length of $OP$ is labelled $r$ .

The axes are drawn as an oblique 3‑D frame: $Z$ is vertical, $Y$ goes to the right, and $X$ comes out toward the lower left. From $L$ , dashed lines drop perpendicularly to each axis, meeting them at the coordinates $x$ , $y$ , $z$ . These three dashed edges, together with the axes, form a rectangular box (a cuboid) whose corner opposite $O$ is $L$ . The coordinates $x$ , $y$ , $z$ are simply the lengths of the sides of that box.

At the origin, three small arcs are drawn — one in each coordinate plane — marking the three direction angles $\alpha$ , $\beta$ , $\gamma$ . These are the angles that the directed line $OL$ makes with the positive $X$ , $Y$ , and $Z$ axes respectively. The figure makes it visually clear that $\alpha$ is the angle between $OL$ and the $X$ -axis, $\beta$ between $OL$ and the $Y$ -axis, and $\gamma$ between $OL$ and the $Z$ -axis.

Note

The dashed cuboid is not just decoration. It shows that the coordinates $x$ , $y$ , $z$ of $L$ are the projections of $OL$ onto the three axes. Because the box is rectangular, each coordinate is the adjacent side in a right triangle whose hypotenuse is $r$ .

From this single picture, the textbook derives the central relation between the direction cosines and the coordinates. In the right triangle formed by $O$ , $P$ , and the foot of the perpendicular from $P$ to the $X$ -axis, the adjacent side is $x$ and the hypotenuse is $r$ . So

\cos\alpha = \frac{x}{r}, \quad \cos\beta = \frac{y}{r}, \quad \cos\gamma = \frac{z}{r}.

These three cosines are called the direction cosines of the directed line $OL$ , and they are denoted by $l$ , $m$ , $n$ :

l = \cos\alpha,\qquad m = \cos\beta,\qquad n = \cos\gamma.

Because $x$ , $y$ , $z$ are the sides of a rectangular box whose space diagonal is $r$ , Pythagoras in three dimensions gives $x^2 + y^2 + z^2 = r^2$ . Dividing through by $r^2$ yields the fundamental identity

l^2 + m^2 + n^2 = 1.

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

This identity is the single most important check for any set of direction cosines. If you ever have three numbers that claim to be direction cosines, their squares must add up to exactly 1.

The figure also sets up the idea of direction ratios. Any three numbers $a$ , $b$ , $c$ that are proportional to $l$ , $m$ , $n$ are called direction ratios of the line. From the geometry, $a$ , $b$ , $c$ can be taken as $x$ , $y$ , $z$ themselves (or any scalar multiple of them), because

\frac{x}{r} = l,\quad \frac{y}{r} = m,\quad \frac{z}{r} = n \quad\Longrightarrow\quad x : y : z = l : m : n.

So the coordinates of any point on the line (other than the origin) give a set of direction ratios. The constant of proportionality $k$ that connects direction ratios to direction cosines is

k = \pm \frac{1}{\sqrt{a^2 + b^2 + c^2}},

and the direction cosines are recovered as …

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