Direction Cosines and Ratios of a Line A line can be defined as a set of points that are equidistant from a given point (the origin). The direction of a line...
A line can be defined as a set of points that are equidistant from a given point (the origin). The direction of a line is determined by the angle it makes with the positive x-axis.
Direction cosines:
The direction cosine (cosθ) of a line with endpoint coordinates (x, y, z) is defined as the ratio of the adjacent side (x) to the hypotenuse (z).
cosθ = x/z
Ratio of a line:
The ratio of the lengths of two segments parallel to the line are equal to the reciprocals of the cosine of the angles that each one makes with the line.
(AB)/(AC) = z/(x)
Examples:
The direction cosines of the line passing through points (1, 2, 3) and (4, 6, 8) are both 0.5. This means that the line is perpendicular to the x-axis.
The ratio of the lengths of the segments AB and BC is 2:3, meaning that AB is twice as long as BC.
The direction cosine of the line passing through points (0, 1, 0) and (1, 2, 3) is 1. This means that the line makes an angle of 45° with the positive x-axis.
Applications of direction cosines and ratios:
Determining the angle of a line.
Calculating the distance from a point to a line.
Identifying the parallel lines to a given line.
Determining the ratio of the lengths of segments parallel to a line