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

Video Transcript

𝐴𝐵𝐶 is an isosceles triangle with a right angle at 𝐵. The points 𝐷, 𝐸, and 𝐹 are the midpoints of the line segments 𝐴𝐶, 𝐴𝐵, and 𝐵𝐶, 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 let’s look at part one. Which of the following planes is an orthonormal coordinate plane? Is it option (A) 𝐴; 𝐸, 𝐷? Option (B) 𝐵; 𝐶, 𝐸. Option (C) plane 𝐵; 𝐹, 𝐸. Option (D) 𝐴; 𝐵, 𝐶. Or option (E) plane 𝐶; 𝐴, 𝐵.

To answer this, we recall that when defining a coordinate plane 𝑂; 𝐼, 𝐽, the first point given is the origin of the coordinate plane. The line from the origin through the second point, that is, the line 𝑂𝐼 forms the 𝑥-axis and the line from the origin through the third point, that is, the line 𝑂𝐽 forms the 𝑦-axis. In part one, we’re looking for an orthonormal coordinate plane, that’s one where the two axes are perpendicular, and the length from the origin to the points 𝐼 and 𝐽. That is, the unit lengths are equal. So let’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 𝐴. The axes are the lines 𝐴𝐸 and 𝐴𝐷. But since our triangle 𝐴𝐵𝐶 is right at angle 𝐵, then angle 𝐷𝐴𝐵 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), 𝐵 is the origin, and the axes are 𝐵𝐶 and 𝐵𝐸. And since our triangle 𝐴𝐵𝐶 has a right angle at 𝐵, then the two axes 𝐵𝐶 and 𝐵𝐸 are indeed perpendicular. So the first criteria, that’s the perpendicularity criteria, is satisfied for option (B). Since our triangle 𝐴𝐵𝐶 is isosceles however, the unit lengths 𝐵𝐶 and 𝐵𝐸 are not equal. The side lengths 𝐵𝐴 and 𝐵𝐶 are equal. However, 𝐵𝐸 is only half of the length of 𝐵𝐴. And that is half of the length of 𝐵𝐶. And since our unit lengths are not equal, we can eliminate option (B).

Now considering option (C), again we have 𝐵 as the origin. Our axes in this case are 𝐵𝐹 and 𝐵𝐸. And we see then since 𝐵 is the origin, our axes are perpendicular. And so our first criteria is satisfied. Now, since 𝐹 and 𝐸 are the midpoints of 𝐵𝐶 and 𝐵𝐴, respectively, and the triangle is isosceles, we know that side lengths 𝐵𝐴 and 𝐵𝐶 are the same. We have 𝐵𝐸 is equal to one over two 𝐵𝐴. 𝐵𝐹 is a half 𝐵𝐶, 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, that’s (D) and (E), in option (D), the origin is at the point 𝐴. So as with option (A), we can discount this, since the angle that 𝐴 is not a right angle and hence the axes are not perpendicular. That is, the axes 𝐴𝐵 and 𝐴𝐶 are not perpendicular. And finally, in option (E), we have 𝐶 as the origin. The axes are 𝐶𝐴 and 𝐶𝐵. 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). That’s the plane 𝐵; 𝐹, 𝐸.

Now moving on to part two, which of the following planes is an orthogonal but not an orthonormal coordinate plane? Option (A) plane 𝐵; 𝐹, 𝐸. Option (B) plane 𝐵; 𝐶, 𝐴. Option (C) plane 𝐷; 𝐵, 𝐶. Option (D) plane 𝐴; 𝐵, 𝐶. Or option (E) plane 𝐵; 𝐹, 𝐴.

Now in an orthogonal plane, the axes are perpendicular. But since the plane we’re looking for is not orthonormal, then our unit lengths will not be equal. So let’s consider our five options. Options (A), (B), and (E) have 𝐵 as the origin. Option (A) has the axes 𝐵𝐹 and 𝐵𝐸. And these are indeed perpendicular, so the first criteria is satisfied. Option (B) has axes 𝐵𝐶 and 𝐵𝐴. And these are perpendicular, so option (B) satisfies the first criteria. And option (E) has axes 𝐵𝐹 and 𝐵𝐴. These also are perpendicular, so our first condition of perpendicularity is satisfied for option (E) also.

Now option (C) has 𝐷 as the origin. And since our triangle 𝐴𝐵𝐶 is isosceles, the point 𝐷 is the perpendicular bisector of 𝐴𝐶. And this means that our axes 𝐷𝐵 and 𝐷𝐶 are indeed perpendicular. So the axes for option (C) are perpendicular. Now considering option (𝐷), we see that the origin is the point 𝐴. With axes 𝐴𝐵 and 𝐴𝐶, 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 don’t want our unit length to be the same so that our coordinate plane is not an orthonormal coordinate plane. So let’s look at these four remaining options. In option (A), the unit lengths are 𝐵𝐹 and 𝐵𝐸. Now 𝐵𝐹 is the midpoint of 𝐵𝐶, so 𝐵𝐹 is one-half 𝐵𝐶. And 𝐵𝐸 is the midpoint of 𝐵𝐴, so 𝐵𝐸 is one-half of 𝐵𝐴. But since our triangle 𝐴𝐵𝐶 is an isosceles triangle, 𝐵𝐴 is equal to 𝐵𝐶. So one over two 𝐵𝐶 is one over two 𝐵𝐴. And this means that 𝐵𝐹 is indeed equal to 𝐵𝐸. 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 𝐵𝐴 and 𝐵𝐶 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 𝐷𝐵 and 𝐷𝐶. And if we consider our triangles, triangle 𝐴𝐵𝐶 is isosceles, so angle 𝐵𝐴𝐶 is equal to angle 𝐵𝐶𝐴, and that’s 45 degrees. And since the line segment 𝐷𝐵 bisects the angle 𝐴𝐵𝐶, which is 90 degrees, we have angle 𝐶𝐵𝐷 is 90 over two, and that’s 45 degrees. So now, we’re looking at triangle 𝐵𝐶𝐷. And this is also an isosceles triangle. And so the side lengths 𝐷𝐵 and 𝐷𝐶 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 𝐵 as our origin and axes 𝐵𝐹 and 𝐵𝐴. And we know already that 𝐵𝐹 is actually one-half of 𝐵𝐴, since 𝐵𝐴 is the same as 𝐵𝐶 and 𝐹 is the midpoint of 𝐵𝐶. 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 𝐵; 𝐹, 𝐴.

So now let’s look at part three. This asks us, which of the following planes is an oblique coordinate plane? Option (A) plane 𝐷; 𝐵, 𝐶. Option (B) plane 𝐵; 𝐶, 𝐷. Option (C) plane 𝐵; 𝐶, 𝐴. Option (D) plane 𝐷; 𝐵, 𝐴. Or option (E) plane 𝐸; 𝐵, 𝐷.

So we’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 𝐷. Its axes are 𝐷𝐵 and 𝐷𝐶. And we’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 𝐵 and the axes are 𝐵𝐶 and 𝐵𝐷. And we’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 𝐵 with axes 𝐵𝐶 and 𝐵𝐴. And these are perpendicular, so we can eliminate option (C). Option (D) has its origin at 𝐷 and has axes 𝐷𝐵 and 𝐷𝐴, which are perpendicular. So we can eliminate option (D). And finally, option (E) has its origin at point 𝐸 and axes 𝐸𝐵 and 𝐸𝐷. 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 𝐵; 𝐹, 𝐸. Our answer for part two is option (E); that’s plane 𝐵; 𝐹, 𝐴. And our answer to part three is option (B), and that’s plane 𝐵; 𝐶, 𝐷.