Direction Ratios 

Direction ratios are a set of three numbers that are proportional to the direction of a line in three-dimensional space. They help define the orientation of the line without fixing its exact length. If a line passes through two points, the differences in their coordinates give the direction ratios. These ratios are useful in vector algebra and 3D geometry for solving problems involving angles, lines, and planes. Direction ratios are often used to find direction cosines and equations of lines.

1.0What Are Direction Ratios?

Direction Ratios (often abbreviated as D.R.s) are a set of three numbers that are proportional to the direction of a line in 3D space.

If a line points in a direction represented by the vector:

$\vec{d}=ai+bj+ck$

Then a, b, c are the direction ratios of the line.

Note: Direction ratios are not unique. Any scalar multiple of a set of direction ratios still represents the same direction.

2.0Visual Representation of Direction Ratios

Here’s a simple illustration:

A line in 3D space with direction ratios (a, b, c), pointing from point A to point B.

3.0Direction Ratios vs Direction Cosines

While direction ratios show the proportion of movement along each axis, direction cosines (denoted as l, m, n) are the cosines of the angles a line makes with the x, y, and z axes, respectively.

If a, b, c are direction ratios, then direction cosines are calculated as:

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

4.0How to Find Direction Ratios

From Two Points

If a line passes through $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, direction ratios are:

$a=x_2-x_1,b=y_2-y_1,c=z_2-z_1$

From a Vector

If a line is parallel to vector $\vec{v}=ai+bj+ck$, then the direction ratios are a, b, c.

5.0Solved Example on Direction Ratios

Example 1: Find the direction ratios of the line passing through the points A(1, 2, 3) and B(4, 6, 8).

Solution:

Direction Ratios = $(x_2-x_1,y_2-y_1,z_2-z_1)$
= (4 − 1, 6 − 2, 8 − 3)
= (3, 4, 5)

Answer: Direction Ratios are 3, 4, 5

Example 2: Find the direction cosines of a line whose direction ratios are 2, -3, 6.

Solution:

Let the direction ratios be a = 2, b = -3, c = 6

Magnitude = $\sqrt{a^2+b^2+c^2}=\sqrt{4+9+36}=\sqrt{49}=7$

Direction Cosines are:

• $l=\frac{2}{7}$
• $m=\frac{-3}{7}$
• $n=\frac{6}{7}$

Answer: Direction cosines are $\frac{2}{7},\frac{-3}{7},\frac{6}{7}$

Example 3: A line has direction ratios 1, 2, 2. Find the equation of the line passing through point (3, -1, 4).

Solution:

The symmetric form of a line passing through point $(x_1,y_1,z_1)$ with D.R.s a, b, c is:

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Substituting values:

$\frac{x-3}{1}=\frac{y+1}{2}=\frac{z-4}{2}$

Answer: Line equation is $\frac{x-3}{1}=\frac{y+1}{2}=\frac{z-4}{2}$

Example 4: A line has direction ratios a, b, c such that it is equally inclined to the coordinate axes. Find the direction ratios and direction cosines of the line.

Solution:

If a line is equally inclined to the x, y, and z axes, then:

$l=m=n\Rightarrow l^2+m^2+n^2=1\Rightarrow 3l^2=1\Rightarrow l=\frac{1}{\sqrt{3}}$

Direction cosines: $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
Direction ratios (any proportional set): 1, 1, 1

Answer: Direction Ratios: 1, 1, 1, Direction Cosines: $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Example 5: Two lines have direction ratios (1, -2, 3) and (2, k, 6). If the lines are perpendicular, find the value of k.

Solution:

For lines to be perpendicular, dot product of direction ratios = 0:

$1\cdot 2+(-2)\cdot k+3\cdot 6=0\Rightarrow 2-2k+18=0\Rightarrow 20-2k=0\Rightarrow k=10$

Answer: k = 10