Question Video: Understanding the Properties of Orthonormal, Orthogonal, and Oblique Coordinate Planes Mathematics

Video Transcript

By considering the given figure, complete the following sentences. The coordinate plane 𝐴; 𝐵, 𝐷 is what? Is it (A) orthonormal, (B) orthogonal but not orthonormal, or (C) oblique?

There are also two further parts to this question which involve different coordinate planes. Let’s begin by recalling the definitions of orthonormal, orthogonal, and oblique coordinate planes. In an oblique coordinate plane, the lines 𝑂𝐼 and 𝑂𝐽, which represent the 𝑥- and 𝑦-axes, respectively, are not perpendicular. In an orthogonal coordinate plane, on the other hand, the 𝑥- and 𝑦-axes are perpendicular. Finally, in an orthonormal coordinate plane, lines 𝑂𝐼 and 𝑂𝐽 representing the 𝑥- and 𝑦-axes are perpendicular. And the length of line segment 𝑂𝐼, the unit length of the 𝑥-axis, is equal to the length of line segment 𝑂𝐽, which is the unit length of the 𝑦-axis.

Let’s now consider the coordinate plane 𝐴; 𝐵, 𝐷 in the figure. When the coordinate plane is written using this notation, the first letter, in this case 𝐴, represents the origin. The line through points 𝐴 and 𝐵 is the 𝑥-axis, and the length of line segment 𝐴𝐵 is the unit length along the 𝑥-axis. In the same way, the line through points 𝐴 and 𝐷 is the 𝑦-axis, and the length of the line segment 𝐴𝐷 is the unit length along the 𝑦-axis. Since the two axes are perpendicular, this rules out option (C). And since 𝐴𝐵𝐹𝐷 is a square, it means the unit lengths in the 𝑥- and 𝑦-directions are equal, so this is an orthonormal coordinate plane. The coordinate plane 𝐴; 𝐵, 𝐷 is orthonormal.

Let’s now consider the second part to this question. This part of the question wants us to consider the coordinate plane 𝐴; 𝐶, 𝐷. Once again, point 𝐴 is the origin. The line through points 𝐴 and 𝐶 is the 𝑥-axis, and the length of line segment 𝐴𝐶 is the unit length along the 𝑥-axis. As with the first part to the question, the line through points 𝐴 and 𝐷 is the 𝑦-axis, and the length of the line segment 𝐴𝐷 is the unit length along the 𝑦-axis. Once again, the two axes are perpendicular, so the coordinate plane is not oblique.

In this question, however, the unit lengths in the 𝑥- and 𝑦-direction are different lengths. In fact, line segment 𝐴𝐶 is double the length of line segment 𝐴𝐷. This means that the coordinate plane is not orthonormal, and we can therefore conclude that the correct answer is option (B) orthogonal. The coordinate plane 𝐴; 𝐶, 𝐷 is orthogonal.

Let’s now consider the third and final part to this question. This time, we need to consider the coordinate plane 𝐵; 𝐶, 𝐷.

The origin this time is located at point 𝐵. The line through points 𝐵 and 𝐶 is the 𝑥-axis, and the unit length in that direction is the length of the line segment 𝐵𝐶. The line through points 𝐵 and 𝐷 is the 𝑦-axis, and the unit length in this direction is equal to the length of line segment 𝐵𝐷. It is clear from the figure that the angle 𝐶𝐵𝐷 is not a right angle. This means that the 𝑥- and 𝑦-axes are not perpendicular, and as such, the coordinate plane is oblique. We can conclude that the coordinate plane 𝐵; 𝐶, 𝐷 is oblique.

We have now completed all three sentences. The coordinate plane 𝐴; 𝐵, 𝐷 is orthonormal, the coordinate plane 𝐴; 𝐶, 𝐷 is orthogonal, and the coordinate plane 𝐵; 𝐶, 𝐷 is oblique.