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

By considering the given figure, complete the following sentences. The coordinate plane (π΄; π΅, π·) is οΌΏ. [A] orthonormal [B] orthogonal but not orthonormal [C] oblique, The coordinate plane (π΄; πΆ, π·) is οΌΏ. [A] orthonormal [B] orthogonal [C] oblique, The coordinate plane (π΅; πΆ, π·) is οΌΏ. [A] orthonormal [B] orthogonal [C] oblique

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### Video Transcript

By considering the given figure, complete the following sentences. The coordinate plane π΄; π΅, π· 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. Letβs begin by recalling the definitions of orthonormal, orthogonal, and oblique coordinate planes. In an oblique coordinate plane, the lines ππΌ and ππ½, which represent the π₯- and π¦-axes, respectively, are not perpendicular. In an orthogonal coordinate plane, on the other hand, the π₯- and π¦-axes are perpendicular. Finally, in an orthonormal coordinate plane, lines ππΌ and ππ½ representing the π₯- and π¦-axes are perpendicular. And the length of line segment ππΌ, the unit length of the π₯-axis, is equal to the length of line segment ππ½, which is the unit length of the π¦-axis.

Letβs now consider the coordinate plane π΄; π΅, π· in the figure. When the coordinate plane is written using this notation, the first letter, in this case π΄, represents the origin. The line through points π΄ and π΅ is the π₯-axis, and the length of line segment π΄π΅ is the unit length along the π₯-axis. In the same way, the line through points π΄ and π· is the π¦-axis, and the length of the line segment π΄π· is the unit length along the π¦-axis. Since the two axes are perpendicular, this rules out option (C). And since π΄π΅πΉπ· is a square, it means the unit lengths in the π₯- and π¦-directions are equal, so this is an orthonormal coordinate plane. The coordinate plane π΄; π΅, π· is orthonormal.

Letβs now consider the second part to this question. This part of the question wants us to consider the coordinate plane π΄; πΆ, π·. Once again, point π΄ is the origin. The line through points π΄ and πΆ is the π₯-axis, and the length of line segment π΄πΆ is the unit length along the π₯-axis. As with the first part to the question, the line through points π΄ and π· is the π¦-axis, and the length of the line segment π΄π· is the unit length along the π¦-axis. Once again, the two axes are perpendicular, so the coordinate plane is not oblique.

In this question, however, the unit lengths in the π₯- and π¦-direction are different lengths. In fact, line segment π΄πΆ is double the length of line segment π΄π·. 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 π΄; πΆ, π· is orthogonal.

Letβs now consider the third and final part to this question. This time, we need to consider the coordinate plane π΅; πΆ, π·.

The origin this time is located at point π΅. The line through points π΅ and πΆ is the π₯-axis, and the unit length in that direction is the length of the line segment π΅πΆ. The line through points π΅ and π· is the π¦-axis, and the unit length in this direction is equal to the length of line segment π΅π·. It is clear from the figure that the angle πΆπ΅π· is not a right angle. This means that the π₯- and π¦-axes are not perpendicular, and as such, the coordinate plane is oblique. We can conclude that the coordinate plane π΅; πΆ, π· is oblique.

We have now completed all three sentences. The coordinate plane π΄; π΅, π· is orthonormal, the coordinate plane π΄; πΆ, π· is orthogonal, and the coordinate plane π΅; πΆ, π· is oblique.