Usually, for threedimensional geometry, we rely on the threedimensional Cartesian plane. Vectors can also be used to describe the lines and the angles they make with the axis. How should we describe a line passing through origin making an angle with different axes? We define them using cosine ratios of the line. While working with threedimensional geometry (used in so many applications such as game designing), it is required to describe the significance of the line present in 3D space.
Let’s assume a line OA which is passing through the origin and makes angles α, β, and γ with x, y, and zaxis respectively, called direction angles. The cosines of these angles cos(α), cos(β), and cos(γ) are called directional cosines of the directed line OA. The figure given below shows the line OA passing through the origin and all the angles it makes with the coordinate axes.
Note: The line can extend in both the directions, and there can be two sets of angles that it makes with coordinate axes. The line makes α, β, and γ and their supplements π  α, π  β and π  γ with the coordinate axes. Now the signs of the directional cosines will be reversed. So, to have unique directional cosines, we choose to make it a directed line.
Let’s say the directional cosines of the lines are l, m, and n.
Direction Ratios
Numbers that are proportional to the direction cosines of the line are called direction ratios of the line. We have assumed l, m, and n as the directional cosines of the lines. Let’s say a, b and c are the directional ratios of the line. Then,
l = k × a, m = k × b and n = k × c
Where k is any constant.
Now we know that,
l^{2} + m^{2} + n^{2} = 1
⇒ k^{2}(a^{2} + b^{2} + c^{2}) = 1
So, now the directional cosines of a line are,
Let’s assume we have a line BC that has directional cosines l, m, and n. Draw a line through the origin that is parallel to this line and let’s take A(x, y, z) a point on this line. We draw a perpendicular from A to the xaxis.
Let OA = r.
So, x = lr and similarly, y = mr and z = nr.
x^{2} + y^{2} + z^{2} = r^{2}(l^{2} + m^{2} + n^{2})
We know that distance of (x, y,z) from origin is r.
x^{2} + y^{2} + z^{2} = r^{2 }
Thus,
l^{2} + m^{2} + n^{2 } = 1
Imagine we have two points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) and a line passes through them. Our goal is to determine the directional cosines of the line that is passing through these points. Let l, m, and n be the directional cosines of the line AB and let’s say the angles they make with the x,y and z axes respectively are α, β, and γ.
In the figure given above, perpendiculars are drawn from the line AB to the XY plane, and they reach the XY plane at CD. Another perpendicular should be drawn from A to BD which meets BD and N. So, now in the triangle ANB, ∠ABN is γ.
In the same way,
So, the directional cosines of the line joining AB are,
Here,
Sample problems based on these concepts,
Example: Find the directional cosines of the x, y, and zaxis.
The xaxis makes an angle of 90°, 90° and 0° with the coordinate axes. We know that the direction cosines are basically cosines of the angle the line makes with different axes.
l = cos(0°) m = cos(90°) and n = cos(90°)
Thus,
l = 1, m = n = 0
Similarly, for yaxis,
l = 0, m = 1 and n = 0
and for zaxis
l = 0, m = 0 and n = 1.
406 videos217 docs164 tests


Explore Courses for Class 10 exam
