Understanding Direction Ratios and Direction Cosines

In three-dimensional geometry, direction ratios and direction cosines are used to describe the direction of a line in space. While direction ratios give a set of three numbers proportional to the direction cosines, the direction cosines are the cosines of the angles made by the line with the positive directions of the coordinate axes.

Let the direction ratios of a line be <A, B, C>. This means that A, B, and C are proportional to the direction cosines \(l\), \(m\), and \(n\).

The relationship is given by:

• \(l = \frac{A}{\sqrt{A^2 + B^2 + C^2}}\)
• \(m = \frac{B}{\sqrt{A^2 + B^2 + C^2}}\)
• \(n = \frac{C}{\sqrt{A^2 + B^2 + C^2}}\)

Here, \(\sqrt{A^2 + B^2 + C^2}\) is the magnitude of the vector whose components are A, B, and C. This term acts as a normalization factor to convert the direction ratios into direction cosines.

Calculating the Sum of Squares of Direction Cosines

A fundamental property of direction cosines is that the sum of the squares of the direction cosines of any line is always equal to 1.

Let's prove this property. We need to find the sum of the squares of the direction cosines, which is \(l^2 + m^2 + n^2\). Using the formulas relating direction ratios to direction cosines:

\(\qquad l^2 = \left(\frac{A}{\sqrt{A^2 + B^2 + C^2}}\right)^2 = \frac{A^2}{A^2 + B^2 + C^2}\)

\(\qquad m^2 = \left(\frac{B}{\sqrt{A^2 + B^2 + C^2}}\right)^2 = \frac{B^2}{A^2 + B^2 + C^2}\)

\(\qquad n^2 = \left(\frac{C}{\sqrt{A^2 + B^2 + C^2}}\right)^2 = \frac{C^2}{A^2 + B^2 + C^2}\)

Now, let's calculate the sum of these squares:

\(\qquad l^2 + m^2 + n^2 = \frac{A^2}{A^2 + B^2 + C^2} + \frac{B^2}{A^2 + B^2 + C^2} + \frac{C^2}{A^2 + B^2 + C^2}\)

Combine the terms since they have a common denominator:

\(\qquad l^2 + m^2 + n^2 = \frac{A^2 + B^2 + C^2}{A^2 + B^2 + C^2}\)

Assuming that the line is not degenerate (meaning \(A^2 + B^2 + C^2 \neq 0\)), the numerator and the denominator are the same non-zero value. Therefore:

\(\qquad l^2 + m^2 + n^2 = 1\)

Applying the Concept to the Given Problem

The question provides the direction ratios of a line as <a + b, b + c, c + a>. Let these be A, B, and C:

• \(A = a + b\)
• \(B = b + c\)
• \(C = c + a\)

We are asked for the sum of the squares of its direction cosines. Based on the property we just discussed and proved, the sum of the squares of the direction cosines (\(l^2 + m^2 + n^2\)) is always equal to 1, regardless of the specific values of the direction ratios (as long as they are not all zero).

Thus, for the given direction ratios <a + b, b + c, c + a>, the sum of the squares of its direction cosines is 1.

Conclusion on the Sum of Squares

The sum of the squares of the direction cosines of any line in three-dimensional space is a fundamental property and is always equal to 1. This is a constant value and does not depend on the specific values of the direction ratios, as long as the line is defined.

Therefore, for a line with direction ratios < a + b, b + c, c + a >, the sum of the squares of its direction cosines is 1.

Revision Table: Direction Cosines and Ratios

Additional Information: Applications of Direction Cosines

Direction cosines are important in various areas of physics and engineering, especially when dealing with vectors and lines in 3D space. Some applications include:

• Vector Representation: A unit vector along a line has its components equal to the direction cosines of the line.
• Angle Between Lines: The cosine of the angle \(\theta\) between two lines with direction cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) is given by \(\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\).
• Geometry Problems: Solving problems involving lines and planes in 3D space, such as finding the projection of a point on a line or the distance between skew lines.
• Physics: Describing the direction of forces, velocities, or fields in 3D.

The property \(l^2 + m^2 + n^2 = 1\) arises from the fact that the vector (l, m, n) is a unit vector, and the square of its magnitude is always 1.