Direction Cosines and Ratios of a Line What is a line? A line is a geometric path consisting of an infinite number of points that move in a straight line...
What is a line?
A line is a geometric path consisting of an infinite number of points that move in a straight line from left to right.
What is the direction of a line?
The direction of a line is defined by its direction cosines. The direction cosine of a line is the ratio of the opposite side length to the adjacent side length in any given coordinate plane.
What are the direction cosines?
Let (x, y) be any coordinate point on the line. The direction cosines are given by:
cos(θ): This measures the angle made between the positive x-axis and the line segment from the origin to the point (x, y). A cosine of 1 corresponds to a right angle, while a cosine of 0 corresponds to a straight angle.
cos(θ'): This measures the angle made between the positive y-axis and the line segment from the origin to the point (x, y).
What is the direction ratio?
The direction ratio of a line is the ratio of the lengths of the perpendicular segments from the origin to any two points on the line.
Example:
Consider the line passing through points A(1, 3) and B(5, 7). The direction cosines of this line would be:
cos(θ) = 5/1 = 1
cos(θ') = 1/5
Therefore, the line is completely determined by its direction cosines, which are both equal to 1. The line also has a direction ratio of 1:5. This means that for any two points on the line, the ratio of the vertical distance to the horizontal distance will always be equal to 1:5.
In summary:
The direction of a line is defined by its direction cosines.
The direction cosines are the ratios of the lengths of the perpendicular segments from the origin to any two points on the line.
The direction ratio of a line is the ratio of the lengths of the perpendicular segments from the origin to any two points on the line