Question

Show that the sum of the squares of the direction cosines of a line through the origin is equal to 1 Hint: Let \((a, b, c)\) be a point on the line at distance 1 from the origin. Write the direction cosines in terms of \((a, b, c)\).

Short answer

The sum of the squares of the direction cosines of a line through the origin is 1: \( l^2 + m^2 + n^2 = 1 \).

Step-by-step solution

- Define Direction Cosines

The direction cosines of a line are the cosines of the angles that the line makes with the positive x, y, and z axes, respectively. Let these direction cosines be denoted by \( l, m, n \). Thus, we have \( l = \cos(\theta_x) \, m = \cos(\theta_y) \, n = \cos(\theta_z) \).

- Express Direction Cosines in Terms of Coordinates

Given a point \((a, b, c)\) on the line at a distance 1 from the origin, we can write the direction cosines in terms of the coordinates of this point. Thus, \( l = \frac{a}{1} = a \, m = \frac{b}{1} = b \, n = \frac{c}{1} = c \).

- Use the Distance Formula

Since the point is at a distance of 1 from the origin, the coordinates satisfy the equation \( a^2 + b^2 + c^2 = 1 \).

- Sum of the Squares of Direction Cosines

Thus, \( l^2 + m^2 + n^2 = 1 \).

Key concepts

Sum of Squares

In geometry, the sum of squares is a central concept, especially when dealing with distance and coordinate calculations. For this exercise, we're focused on the sum of squares of the direction cosines. Direction cosines, represented by the symbols \( l, m, \) and \( n \), measure the angles a line makes with the coordinate axes. These are fundamental in determining the orientation of a line.

To find the sum of the squares of these cosines, each cosine is squared and then added together. Based on the given exercise, if a point \((a, b, c)\) lies on a line at a distance of 1 from the origin, we find that:

\( l = a, \ m = b, \ n = c \)

Since the line passes through the origin and the distance is 1, the equation \(a^2 + b^2 + c^2 = 1\) holds true. Therefore, we can state that:

\( l^2 + m^2 + n^2 = a^2 + b^2 + c^2 = 1 \).

This simple derivation shows the foundational importance of the sum of squares in such contexts.

Distance Formula

The distance formula is a crucial tool in coordinate geometry. It helps in finding the distance between two points in a space. The formula is given by:

\[ d = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 } \]

In our exercise, we consider a point \((a, b, c)\) on a line at a distance of 1 from the origin \((0, 0, 0)\). By substituting the coordinates into the distance formula, we have:

\[ 1 = \sqrt{ (a-0)^2 + (b-0)^2 + (c-0)^2 } \]

This equation simplifies to:

\[ 1 = \sqrt{ a^2 + b^2 + c^2 } \]

Squaring both sides results in:

\[ a^2 + b^2 + c^2 = 1 \]

This confirms that the sum of the squares of the coordinates of the point is 1, and directly relates to the exercise's goal of proving the sum of the squares of the direction cosines.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a major branch of mathematics that uses algebra to describe geometric principles. It allows us to precisely define geometric shapes and their properties using coordinates and equations.

In the given exercise, coordinate geometry involves points, distances, and direction cosines. The point \((a, b, c)\) describes a specific location in a 3D plane, and its relation to the origin can be analyzed using coordinates. By representing the direction cosines \( l, m, \) and \( n \) in terms of \(a, b, \) and \(c\), we utilize the fundamental principles of coordinate geometry to understand the line's orientation in space.

Through these coordinates, we see the power of coordinate geometry in providing a systematic way to prove that the sum of the squares of the direction cosines is always 1 for any line passing through the origin. This systematization is key in solving various problems in mathematics and physics, grounding abstract concepts in a concrete, calculable manner.