Question Video: Identifying Orthonormal, Orthogonal, and Oblique Coordinate Planes | Nagwa Question Video: Identifying Orthonormal, Orthogonal, and Oblique Coordinate Planes | Nagwa

# Question Video: Identifying Orthonormal, Orthogonal, and Oblique Coordinate Planes Mathematics

π΄π΅πΆ is an isosceles triangle with a right angle at π΅. The points π·, πΈ, and πΉ are the midpoints of the line segments π΄πΆ, π΄π΅, and π΅πΆ respectively. Part 1: Which of the following planes is an orthonormal coordinate plane? [A] (π΄; πΈ, π·) [B] (π΅; πΆ, πΈ) [C] (π΅; πΉ, πΈ) [D] (π΄; π΅, πΆ) [E] (πΆ; π΄, π΅) Part 2: Which of the following planes is an orthogonal but not an orthonormal coordinate plane? [A] (π΅; πΉ, πΈ) [B] (π΅; πΆ, π΄) [C] (π·; π΅, πΆ) [D] (π΄; π΅, πΆ) [E] (π΅; πΉ, π΄) Part 3: Which of the following planes is an oblique coordinate plane? [A] (π·; π΅, πΆ) [B] (π΅; πΆ, π·) [C] (π΅; πΆ, π΄) [D] (π·; π΅, π΄) [E] (πΈ; π΅, π·)

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### Video Transcript

π΄π΅πΆ is an isosceles triangle with a right angle at π΅. The points π·, πΈ, and πΉ are the midpoints of the line segments π΄πΆ, π΄π΅, and π΅πΆ, respectively. There are three parts to this question. Part one asks us, which of the given planes is an orthonormal coordinate plane? Part two asks us, which of the following planes is an orthogonal but not an orthonormal coordinate plane? And part three asks us, which of the following planes is an oblique coordinate plane?

So letβs look at part one. Which of the following planes is an orthonormal coordinate plane? Is it option (A) π΄; πΈ, π·? Option (B) π΅; πΆ, πΈ. Option (C) plane π΅; πΉ, πΈ. Option (D) π΄; π΅, πΆ. Or option (E) plane πΆ; π΄, π΅.

To answer this, we recall that when defining a coordinate plane π; πΌ, π½, the first point given is the origin of the coordinate plane. The line from the origin through the second point, that is, the line ππΌ forms the π₯-axis and the line from the origin through the third point, that is, the line ππ½ forms the π¦-axis. In part one, weβre looking for an orthonormal coordinate plane, thatβs one where the two axes are perpendicular, and the length from the origin to the points πΌ and π½. That is, the unit lengths are equal. So letβs begin by going through each of the given options to see which satisfy the perpendicularity criterion.

In option (A), our origin is the point π΄. The axes are the lines π΄πΈ and π΄π·. But since our triangle π΄π΅πΆ is right at angle π΅, then angle π·π΄π΅ cannot be a right angle. It must be less than 90 degrees, and therefore our axes cannot be perpendicular. And so we can eliminate option (A). In option (B), π΅ is the origin, and the axes are π΅πΆ and π΅πΈ. And since our triangle π΄π΅πΆ has a right angle at π΅, then the two axes π΅πΆ and π΅πΈ are indeed perpendicular. So the first criteria, thatβs the perpendicularity criteria, is satisfied for option (B). Since our triangle π΄π΅πΆ is isosceles however, the unit lengths π΅πΆ and π΅πΈ are not equal. The side lengths π΅π΄ and π΅πΆ are equal. However, π΅πΈ is only half of the length of π΅π΄. And that is half of the length of π΅πΆ. And since our unit lengths are not equal, we can eliminate option (B).

Now considering option (C), again we have π΅ as the origin. Our axes in this case are π΅πΉ and π΅πΈ. And we see then since π΅ is the origin, our axes are perpendicular. And so our first criteria is satisfied. Now, since πΉ and πΈ are the midpoints of π΅πΆ and π΅π΄, respectively, and the triangle is isosceles, we know that side lengths π΅π΄ and π΅πΆ are the same. We have π΅πΈ is equal to one over two π΅π΄. π΅πΉ is a half π΅πΆ, and these are equal. And so for option (C), our second criteria is also satisfied. The unit lengths are equal. And so the coordinate plane defined in option (C) is an orthonormal coordinate plane.

If we look at our remaining options, thatβs (D) and (E), in option (D), the origin is at the point π΄. So as with option (A), we can discount this, since the angle that π΄ is not a right angle and hence the axes are not perpendicular. That is, the axes π΄π΅ and π΄πΆ are not perpendicular. And finally, in option (E), we have πΆ as the origin. The axes are πΆπ΄ and πΆπ΅. And so the angle between them cannot be 90 degrees. The axes are not perpendicular. Hence, we can discount option (E). And hence, the answer to part one of the question, which of the planes is orthonormal?, is option (C). Thatβs the plane π΅; πΉ, πΈ.

Now moving on to part two, which of the following planes is an orthogonal but not an orthonormal coordinate plane? Option (A) plane π΅; πΉ, πΈ. Option (B) plane π΅; πΆ, π΄. Option (C) plane π·; π΅, πΆ. Option (D) plane π΄; π΅, πΆ. Or option (E) plane π΅; πΉ, π΄.

Now in an orthogonal plane, the axes are perpendicular. But since the plane weβre looking for is not orthonormal, then our unit lengths will not be equal. So letβs consider our five options. Options (A), (B), and (E) have π΅ as the origin. Option (A) has the axes π΅πΉ and π΅πΈ. And these are indeed perpendicular, so the first criteria is satisfied. Option (B) has axes π΅πΆ and π΅π΄. And these are perpendicular, so option (B) satisfies the first criteria. And option (E) has axes π΅πΉ and π΅π΄. These also are perpendicular, so our first condition of perpendicularity is satisfied for option (E) also.

Now option (C) has π· as the origin. And since our triangle π΄π΅πΆ is isosceles, the point π· is the perpendicular bisector of π΄πΆ. And this means that our axes π·π΅ and π·πΆ are indeed perpendicular. So the axes for option (C) are perpendicular. Now considering option (π·), we see that the origin is the point π΄. With axes π΄π΅ and π΄πΆ, we know that the angle between them cannot be 90 degrees. And so in this case, we can eliminate option (D).

So we still have options (A), (B), (C), and (E) to consider. We know that we donβt want our unit length to be the same so that our coordinate plane is not an orthonormal coordinate plane. So letβs look at these four remaining options. In option (A), the unit lengths are π΅πΉ and π΅πΈ. Now π΅πΉ is the midpoint of π΅πΆ, so π΅πΉ is one-half π΅πΆ. And π΅πΈ is the midpoint of π΅π΄, so π΅πΈ is one-half of π΅π΄. But since our triangle π΄π΅πΆ is an isosceles triangle, π΅π΄ is equal to π΅πΆ. So one over two π΅πΆ is one over two π΅π΄. And this means that π΅πΉ is indeed equal to π΅πΈ. This means that the unit lengths for option (A) are equal. So we can eliminate option (A). In fact, by the same logic, we can eliminate option (B). Since sides π΅π΄ and π΅πΆ are the equal sides of an isosceles triangle, so option (B) does not satisfy our second criteria. And we can eliminate option (B).

Next, looking at option (C), our unit lengths are the lengths π·π΅ and π·πΆ. And if we consider our triangles, triangle π΄π΅πΆ is isosceles, so angle π΅π΄πΆ is equal to angle π΅πΆπ΄, and thatβs 45 degrees. And since the line segment π·π΅ bisects the angle π΄π΅πΆ, which is 90 degrees, we have angle πΆπ΅π· is 90 over two, and thatβs 45 degrees. So now, weβre looking at triangle π΅πΆπ·. And this is also an isosceles triangle. And so the side lengths π·π΅ and π·πΆ are in fact equal. This means that the unit lengths for option (C) are equal. Hence, we can eliminate option (C), since option (C) represents an orthonormal coordinate plane.

So now finally considering option (E), we have π΅ as our origin and axes π΅πΉ and π΅π΄. And we know already that π΅πΉ is actually one-half of π΅π΄, since π΅π΄ is the same as π΅πΆ and πΉ is the midpoint of π΅πΆ. So for option (E), our unit lengths are not the same and the plane represented by (E) is not orthonormal, but it is orthogonal. So the answer to part two is option (E), the plane π΅; πΉ, π΄.

So now letβs look at part three. This asks us, which of the following planes is an oblique coordinate plane? Option (A) plane π·; π΅, πΆ. Option (B) plane π΅; πΆ, π·. Option (C) plane π΅; πΆ, π΄. Option (D) plane π·; π΅, π΄. Or option (E) plane πΈ; π΅, π·.

So weβre looking for an oblique coordinate plane that is a coordinate plane whose axes are not perpendicular. We see option (A) has its origin at the point π·. Its axes are π·π΅ and π·πΆ. And weβve seen already that these are actually perpendicular. So we can eliminate option (A). For option (B) on the other hand, the origin is at π΅ and the axes are π΅πΆ and π΅π·. And weβve seen already that the angle between these two is not 90 degrees. Hence, the axes for option (B) are not perpendicular. So (B) does represent an oblique coordinate plane.

Option (C) has its origin at π΅ with axes π΅πΆ and π΅π΄. And these are perpendicular, so we can eliminate option (C). Option (D) has its origin at π· and has axes π·π΅ and π·π΄, which are perpendicular. So we can eliminate option (D). And finally, option (E) has its origin at point πΈ and axes πΈπ΅ and πΈπ·. And since these axes are perpendicular, we can eliminate option (E). And of our options, only plane (B) is an oblique coordinate plane.

Our answer to part one is option (C), plane π΅; πΉ, πΈ. Our answer for part two is option (E); thatβs plane π΅; πΉ, π΄. And our answer to part three is option (B), and thatβs plane π΅; πΆ, π·.

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