In Class 12 mathematics, direction ratios are a set of numbers that describe the orientation or direction of a line or vector in three-dimensional space. They are essentially the components of a vector along the x-axis, y-axis, and z-axis, representing the proportional changes along each coordinate axis for a given line or vector.
Understanding Direction Ratios
Imagine a line segment or a vector originating from the origin. Its direction ratios are any three numbers (a, b, c) that are proportional to the actual displacements along the X, Y, and Z axes. This means if you move a units along the x-axis, b units along the y-axis, and c units along the z-axis, you will be moving in a direction parallel to the given line or vector.
For example, if a vector is given as $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$, then a, b, c are its direction ratios.
Direction Ratios vs. Direction Cosines
It's crucial to differentiate direction ratios from direction cosines, which are also used to describe direction.
- Direction Cosines ($l, m, n$): These are the cosines of the angles (α, β, γ) a line makes with the positive directions of the x-axis, y-axis, and z-axis, respectively. So, $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$. A key property is that $l^2 + m^2 + n^2 = 1$.
- Direction Ratios ($a, b, c$): These are any three numbers that are proportional to the direction cosines. This means $a = kl$, $b = km$, $c = kn$ for some non-zero scalar $k$.
This proportionality implies that a line can have infinitely many sets of direction ratios (e.g., (1, 2, 3), (2, 4, 6), (-1, -2, -3) all represent the same direction), whereas it has a unique set of direction cosines.
Relationship Between Direction Ratios and Direction Cosines
If a, b, c are the direction ratios of a line, then its direction cosines l, m, n can be calculated using the following formulas:
- $l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}$
- $m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}$
- $n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$
The term $\sqrt{a^2 + b^2 + c^2}$ represents the magnitude of a vector whose components are a, b, c.
How to Find Direction Ratios
Direction ratios can be determined in various ways depending on the given information:
-
From Two Points:
If a line passes through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, its direction ratios are given by:
$(x_2 - x_1)$, $(y_2 - y_1)$, $(z_2 - z_1)$
or
$(x_1 - x_2)$, $(y_1 - y_2)$, $(z_1 - z_2)$. -
From a Vector Equation:
If a line is represented by a vector equation $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ is the vector parallel to the line, then $b_1, b_2, b_3$ are the direction ratios of the line. -
From a Cartesian Equation of a Line:
For a line given by the equation $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$, the denominators a, b, c represent the direction ratios of the line.
Important Properties and Applications
- Collinearity: Three points $A, B, C$ are collinear if the direction ratios of line $AB$ are proportional to the direction ratios of line $BC$.
- Parallel Lines: Two lines are parallel if their direction ratios are proportional.
- Perpendicular Lines: If $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are the direction ratios of two lines, they are perpendicular if $a_1a_2 + b_1b_2 + c_1c_2 = 0$.
- Angle Between Lines: Direction ratios are crucial for calculating the angle between two lines using the formula for the cosine of the angle.
- Equations of Lines and Planes: Direction ratios form an integral part of the vector and Cartesian equations of lines and planes in 3D geometry.
Example
Problem: Find the direction ratios of the line passing through the points $A(1, 2, 3)$ and $B(4, 5, 6)$.
Solution:
Using the formula for direction ratios from two points $(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)$:
Direction ratios are $(4 - 1), (5 - 2), (6 - 3)$.
So, the direction ratios are $(3, 3, 3)$.
We could also use $(1-4), (2-5), (3-6)$, which gives $(-3, -3, -3)$. Both sets represent the same direction, differing only by a scalar multiple.
Summary Table: Direction Ratios vs. Direction Cosines
| Feature | Direction Ratios ($a, b, c$) | Direction Cosines ($l, m, n$) |
|---|---|---|
| Definition | Components proportional to displacement along axes. | Cosines of angles made with positive coordinate axes. |
| Uniqueness | Not unique (any scalar multiple is also a valid set). | Unique for a given direction. |
| Value Range | Can be any three real numbers (not all zero). | Values range from -1 to 1. |
| Key Property | Represented by any set of numbers proportional to $l, m, n$. | $l^2 + m^2 + n^2 = 1$. |
| Calculation | Found from difference of coordinates or vector components. | Calculated from direction ratios or angles with axes. |
Direction ratios are a fundamental concept in three-dimensional geometry, providing a simple yet powerful way to express the orientation of lines and vectors, which is essential for solving various problems in vector algebra and coordinate geometry. To learn more about related concepts, you can explore resources on vector algebra and 3D coordinate geometry.
