Understanding Direction Ratios in 3D Geometry

In 3D geometry, direction ratios (DRs) are a set of three numbers that are proportional to the direction cosines of a line. They provide a way to represent the direction of a line without normalizing the length of the vector to 1, as is done with direction cosines.

What are Direction Ratios?

Direction ratios are simply the coefficients of the vector representing the direction of a line. If a line is represented by the vector $\vec{a} = a\hat{i} + b\hat{j} + c\hat{k}$, then the direction ratios of the line are $a$, $b$, and $c$.

Relation between Direction Ratios and Direction Cosines

Direction cosines are the cosines of the angles that a line makes with the positive directions of the coordinate axes. If $\alpha$, $\beta$, and $\gamma$ are the angles that a line makes with the $x$-, $y$-, and $z$-axes respectively, then the direction cosines are $\cos\alpha$, $\cos\beta$, and $\cos\gamma$.

The direction ratios are proportional to the direction cosines. If $l$, $m$, and $n$ are the direction cosines, then the direction ratios $a$, $b$, and $c$ can be written as:

$$ a = kl $$ $$ b = km $$ $$ c = kn $$

where $k$ is a non-zero scalar.

Properties of Direction Ratios

Here are some important properties of direction ratios:

• They are not unique for a given line. If $(a, b, c)$ are the direction ratios of a line, then any scalar multiple $(ka, kb, kc)$, where $k$ is a non-zero real number, are also direction ratios of the same line.
• They can be positive, negative, or zero.
• They can be used to find the angle between two lines.

Formulas Involving Direction Ratios

Angle Between Two Lines

If two lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the cosine of the angle $\theta$ between them is given by:

$$ \cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} $$

Direction Ratios of a Line Joining Two Points

If a line joins two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, the direction ratios of the line are given by the differences in the coordinates:

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

Examples

Example 1: Finding Direction Ratios

Given a line with direction cosines $l = \frac{1}{\sqrt{3}}$, $m = \frac{1}{\sqrt{3}}$, $n = \frac{1}{\sqrt{3}}$, find the direction ratios.

Since direction ratios are proportional to direction cosines, we can choose $k = \sqrt{3}$ to get integers:

$$ a = k \cdot l = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 $$ $$ b = k \cdot m = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 $$ $$ c = k \cdot n = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 $$

So, the direction ratios are $(1, 1, 1)$.

Example 2: Angle Between Two Lines

Find the angle between the lines with direction ratios $(1, 2, 3)$ and $(4, -1, 2)$.

Using the formula for the angle between two lines:

$$ \cos\theta = \frac{1 \cdot 4 + 2 \cdot (-1) + 3 \cdot 2}{\sqrt{1^2 + 2^2 + 3^2} \cdot \sqrt{4^2 + (-1)^2 + 2^2}} $$ $$ \cos\theta = \frac{4 - 2 + 6}{\sqrt{1 + 4 + 9} \cdot \sqrt{16 + 1 + 4}} $$ $$ \cos\theta = \frac{8}{\sqrt{14} \cdot \sqrt{21}} $$ $$ \theta = \cos^{-1}\left(\frac{8}{\sqrt{294}}\right) $$

Table: Differences and Important Points

Property	Direction Ratios	Direction Cosines
Definition	Proportional to the vector components	Cosines of the angles with axes
Uniqueness	Not unique (scalar multiples)	Unique for a given line
Values	Can be any real number	Range from -1 to 1
Usage	Used to represent direction	Used to find angles with axes

Direction ratios are a fundamental concept in 3D geometry, especially when dealing with lines and planes. Understanding how to work with them is crucial for solving problems in vector algebra and analytical geometry.