### 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. Letβs begin by recalling the
definitions of orthonormal, orthogonal, and oblique coordinate planes. In an oblique coordinate plane, the
lines ππΌ and ππ½, 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 ππΌ and ππ½ representing the π₯- and π¦-axes are
perpendicular. And the length of line segment
ππΌ, the unit length of the π₯-axis, is equal to the length of line segment ππ½,
which is the unit length of the π¦-axis.

Letβs 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΅πΉπ· 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.

Letβs now consider the second part
to this question. This part of the question wants us
to consider the coordinate plane π΄; πΆ, π·. Once again, point π΄ is the
origin. The line through points π΄ and πΆ
is the π₯-axis, and the length of line segment π΄πΆ 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΄πΆ 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΄; πΆ, π· is
orthogonal.

Letβs now consider the third and
final part to this question. This time, we need to consider the
coordinate plane π΅; πΆ, π·.

The origin this time is located at
point π΅. The line through points π΅ and πΆ
is the π₯-axis, and the unit length in that direction is the length of the line
segment π΅πΆ. 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 πΆπ΅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΅; πΆ, π· is oblique.

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