Direction Cosines And Direction Ratios Of A Line

We are given the following two equations that describe the direction cosines of two lines: 1) 2) We are asked to find the angle between these two lines. The direction cosines of the lines are represented by , , and , where: - is the cosine of the angle between the line and the -axis, - is the cosine of the angle between the line and the -axis, and - is the cosine of the angle between the line and the -axis.
Step 1: Use the first equation to express From the first equation , we can express in terms of and :
Step 2: Substitute into the second equation Substitute this expression for into the second equation : Expanding the terms: Simplifying:
Step 3: Factor the equation Factor the equation: Thus, either or .
Step 4: Solve for the angle between the lines Let's consider the case where . Substituting into : Thus, the direction cosines of the lines are: The formula for the angle between two lines in terms of their direction cosines and is given by: Substituting and : For , we get , and thus . Finally, we find that the angle is: Thus, the angle between the lines is .