OBLIQUE LINES: CHARACTERISTICS, EQUATIONS AND EXAMPLES - MATHS - 2025

The oblique lines are those that are inclined, either relative to a flat surface or other line indicating a particular address. As an example, consider the three lines drawn in a plane that appear in the following figure.

We know their respective relative positions because we compare them to a reference line, which is usually the x-axis denoting the horizontal.

Figure 1. Vertical, horizontal and oblique lines in the same plane. Source: F. Zapata.

In this way, choosing the horizontal as a reference, the line on the left is vertical, the one in the center is horizontal and the one on the right is oblique, since it is inclined with respect to the daily reference lines.

Now, the lines that are on the same plane, such as the surface of the paper or the screen, occupy different positions relative to each other, depending on whether they intersect or not. In the first case they are secant lines, while in the second, they are parallel.

On the other hand, the secant lines can be oblique lines or perpendicular lines. In both cases, the slopes of the lines are different, but the oblique lines form angles α and β between them, different from 90º, while the angles determined by the perpendicular lines are always 90º.

The following figure summarizes these definitions:

Figure 2. Relative positions between lines: parallel, oblique and perpendicular differ in the angle they form with each other. Source: F. Zapata.

Equations

To know the relative positions of the lines in the plane, it is necessary to know the angle between them. Note that the lines are:

Parallel: if they have the same slope (the same direction) and never intersect, therefore their points are equidistant.

Coincidents: when all its points coincide and therefore have the same slope, but the distance between its points is zero.

Dryers: if their slopes are different, the distance between their points varies and the intersection is a single point.

So one way to know if two lines in the plane are secant or parallel is through their slope. The criteria of parallelism and perpendicularity of the lines are the following:

If, knowing the slopes of two lines in the plane, none of the above criteria is met, we conclude that the lines are oblique. Knowing two points on a line, the slope is calculated immediately, as we will see in the next section.

You can find out if two lines are secant or parallel by finding their intersection, solving the system of equations they form: if there is a solution, they are secant, if there is no solution, they are parallel, but if the solutions are infinite, the lines are coincident.

However, this criterion does not inform us about the angle between these lines, even if they intersect.

To know the angle between the lines, we need two vectors u and v that belong to each of them. Thus it is possible to know the angle they form by means of the scalar product of the vectors, defined in this way:

u • v = uvcos α

Equation of the line in the plane

A line in the Cartesian plane can be represented in several ways, such as:

- Slope-intercept form: if m is the slope of the line and b is the intersection of the line with the vertical axis, the equation of the line is y = mx + b.

- General equation of the line: Ax + By + C = 0, where m = A / B is the slope.

In the Cartesian plane, vertical and horizontal lines are particular cases of the equation of the line.

- Vertical lines: x = a

- Horizontal lines: y = k

Figure 3. On the left the vertical line x = 4 and the horizontal line y = 6. On the right an example of an oblique line. Source: F. Zapata.

In the examples in figure 3, the vertical red line has equation x = 4, while the line parallel to the x-axis (blue) has equation y = 6. As for the line on the right, we see that it is oblique and to find its equation we use the points highlighted in the figure: (0,2) and (4,0) in this way:

The cut of this line with the vertical axis is y = 2, as can be seen from the graph. With this information:

Determining the angle of inclination with respect to the x-axis is easy. I feel that:

Therefore the positive angle from the x axis to the line is: 180º - 26.6º = 153.4º

Examples of oblique lines

Figure 4. Examples of oblique lines. Source: fencers Ian Patterson. Pisa's leaning tower. Pixabay.

Oblique lines appear in many places, it is a matter of paying attention to find them in architecture, sports, electrical wiring, plumbing and in many more places. In nature the oblique lines are also present, as we will see below:

Rays of light

Sunlight travels in a straight line, but the round shape of the Earth affects how sunlight hits the surface.

In the image below we can clearly see that the sun's rays strike perpendicularly in tropical regions, but instead reach the surface obliquely in temperate regions and at the poles.

This is why the sun's rays travel a longer distance through the atmosphere and also the heat spreads over a larger surface (see figure). The result is that the areas near the poles are colder.

Figure 5. The sun's rays fall obliquely in the temperate zones and the poles, instead they are more or less perpendicular in the tropics. Source: Wikimedia Commons.

Lines that are not in the same plane

When two lines are not in the same plane, they can still be oblique or warped, as they are also known. In this case, their director vectors are not parallel, but since they do not belong to the same plane, these lines do not intersect.

For example, the lines in figure 6 right are clearly in different planes. If you look at them from above, you can see that they do intersect, but they do not have a point in common. On the right we see the wheels of the bicycle, whose spokes seem to cross when viewed from the front.

Figure 6. Oblique lines belonging to different planes. Source: left F. Zapata, right Pixabay.

References

1. Geometry. Director vector of a line. Recovered from: juanbragado.es.
2. Larson, R. 2006. Calculus with Analytical Geometry. 8th. Edition. McGraw Hill.
3. Mathematics is a game. Lines and Angles. Recovered from: juntadeandalucia.es.
4. Straight lines that intersect. Recovered from: profesoraltuna.com.
5. Villena, M. Analytical Geometry in R3. Recovered from: dspace.espol.edu.ec.