Angle in Terms of Direction Ratios

In 3D geometry, direction ratios are a set of three numbers that are proportional to the direction cosines of a line. These ratios provide a way to represent the orientation of a line in three-dimensional space. When we talk about the angle between two lines, we can use the direction ratios to find this angle.

Understanding Direction Ratios

Direction ratios (often denoted as a, b, and c) are simply numbers that are proportional to the direction cosines (l, m, n) of a line. The direction cosines are the cosines of the angles that the line makes with the positive directions of the x, y, and z-axes, respectively.

The relationship between direction ratios and direction cosines is given by:

$$ \begin{align*} l &= \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \ m &= \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \ n &= \frac{c}{\sqrt{a^2 + b^2 + c^2}}. \end{align*} $$

Angle Between Two Lines

The angle θ between two lines with direction ratios a_1, b_1, c_1 and a_2, b_2, c_2 can be found using the dot product of their direction vectors. The formula for the cosine of the angle is:

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

Table of Differences and Important Points

Aspect	Direction Ratios (DR)	Direction Cosines (DC)
Definition	Proportional to the components of the direction vector of a line.	Cosines of the angles made with the positive x, y, and z axes.
Representation	Any three numbers (a, b, c) that are proportional to the DC.	Unique for a given line and are denoted by (l, m, n).
Normalization	Not necessarily normalized.	Always normalized such that ( l^2 + m^2 + n^2 = 1 ).
Calculation of Angles	Used to calculate the angle between two lines using the dot product.	Directly used to find the angle with respect to the axes.

Examples

Example 1: Finding the Angle Between Two Lines

Suppose we have two lines with direction ratios L1: (1, 2, 3) and L2: (4, -1, 2). To find the angle between these two lines, we use the formula for the cosine of the angle:

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

From this, we can calculate the angle θ.

Example 2: Direction Cosines from Direction Ratios

Given direction ratios a = 3, b = -6, c = 2, find the direction cosines. First, we normalize the direction ratios:

$$ l = \frac{3}{\sqrt{3^2 + (-6)^2 + 2^2}}, \quad m = \frac{-6}{\sqrt{3^2 + (-6)^2 + 2^2}}, \quad n = \frac{2}{\sqrt{3^2 + (-6)^2 + 2^2}}. $$

Simplifying, we get:

$$ l = \frac{3}{7}, \quad m = \frac{-6}{7}, \quad n = \frac{2}{7}. $$

These are the direction cosines for the given direction ratios.

Example 3: Using Direction Ratios to Find the Angle with Axes

If a line has direction ratios a = 4, b = 4, c = 2, we can find the angle it makes with the x-axis by finding the direction cosine l:

$$ l = \frac{4}{\sqrt{4^2 + 4^2 + 2^2}} = \frac{4}{\sqrt{36}} = \frac{2}{3}. $$

The angle θ with the x-axis is then given by:

$$ \theta = \cos^{-1}\left(\frac{2}{3}\right). $$

In conclusion, understanding the concept of direction ratios and how they relate to direction cosines is crucial for solving problems involving angles between lines in 3D geometry. The formulas and examples provided should help solidify this understanding and prepare students for exams on this topic.