Video Transcript
In this video, we’ll learn how to
define the different types of coordinate planes and the coordinates of a point and
place them on the plane. Coordinate planes are used in
particular for specifying an object’s location using its coordinates in the
plane. On Earth, we commonly use the
geographic coordinate system based on latitude and longitude. And in geometry, the most familiar
coordinate plane is called the orthonormal coordinate plane. And that’s where the axes are
perpendicular and the spacings are equal. In this video, we’ll look at
coordinate planes of this type and also some alternative coordinate planes.
Let’s begin by giving a general
definition for a coordinate plane. A coordinate plane is formed by any
three noncollinear points 𝑂; 𝐼, 𝐽, where 𝑂 is the origin, the line 𝑂𝐼 is the
𝑥-axis with its positive direction and the direction of 𝑂𝐼, and line 𝑂𝐽 is the
𝑦-axis with its positive direction in the direction of 𝑂𝐽. The length of the line segment 𝑂𝐼
is the unit length of the 𝑥-axis. And the length of line segment 𝑂𝐽
is the unit length of the 𝑦-axis. And from this definition, we see
that the orthonormal coordinate plane is a special coordinate plane with its 𝑥-axis
𝑂𝐼 perpendicular to its 𝑦-axis 𝑂𝐽 and where the unit lengths 𝑂𝐼 and 𝑂𝐽 are
equal.
Now back to the general case, let’s
take three random noncollinear points 𝑂, 𝐼, and 𝐽 positioned as shown. If we want to use them to form a
coordinate plane, we draw the lines 𝑂𝐼 and 𝑂𝐽 to form the 𝑥- and 𝑦-axes,
respectively, and create a grid with lines parallel to both axes and spaced by the
unit lengths 𝑂𝐼 and 𝑂𝐽. The coordinate plane shown is
called an oblique coordinate plane. And this is characterized by the
fact that the axes are not perpendicular and that the grid forms parallelograms. Another type of coordinate plane is
an orthogonal coordinate plane. In this case, the axes 𝑂𝐼 and
𝑂𝐽, that’s the 𝑥- and 𝑦-axes, are perpendicular, although the unit lengths 𝑂𝐼
and 𝑂𝐽 may not be the same. And this grid forms rectangles. And finally, an orthonormal
coordinate plane is one where the axes 𝑂𝐼 and 𝑂𝐽 are perpendicular and where the
unit lengths 𝑂𝐼 and 𝑂𝐽 are equal and the grid forms squares.
It’s worth noting also that we
normally represent coordinate planes with a horizontal 𝑥-axis for convenience. So now, let’s look at an example of
how we might use our definitions to identify different types of coordinate
planes.
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.
So now that we have defined our
three types of coordinate plane, let’s now see how we define coordinates in a
coordinate plane. Given a coordinate plane 𝑂; 𝐼,
𝐽, the position of any point 𝑀 in the plane is described by its coordinates. These are denoted 𝑥 sub 𝑀 and 𝑦
sub 𝑀. And that’s where 𝑥 sub 𝑀 is the
real number on the 𝑥-axis at the point of intersection of the line parallel to the
𝑦-axis and through 𝑀. And 𝑦 𝑀 is the real number on the
𝑦-axis at the point of intersection of the line parallel to the 𝑥-axis and through
𝑀.
So if we consider, for example, an
oblique coordinate plane, the point 𝑀 has coordinates 𝑥 sub 𝑀 and 𝑦 sub 𝑀. And we note by definition in the
plane 𝑂; 𝐼, 𝐽, the point 𝐼 has coordinates one, zero and the point 𝐽 has
coordinates zero, one. Let’s look at a final example in an
orthonormal coordinate plane.
𝐴 and 𝐵 are two points in an
orthonormal coordinate plane with a positive horizontal 𝑥-axis pointing to the
right and a positive vertical 𝑦-axis pointing up. The unit lengths of the axes are
given by the grid. If the coordinates of 𝐴 are one,
two, what are the coordinates of 𝐵?
Our points 𝐴 and 𝐵 are in an
orthonormal coordinate plane. And that’s a plane whose unit
lengths are given by the grid. Now remember, an orthonormal
coordinate plane is plane 𝑂; 𝐼, 𝐽 whose axes 𝑂𝐼 and 𝑂𝐽 are perpendicular,
that’s the 𝑥- and 𝑦-axes, and whose unit lengths 𝑂𝐼 and 𝑂𝐽 are equal. To find the coordinates of the
point 𝐵, we must first determine the position of the origin of the coordinate
plane. We can do this using the given
point 𝐴. We know that 𝐴 has coordinates
one, two. And this means that 𝐴 sits one
unit to the right of the origin and two units up from the origin. And so working backwards, the
origin is one unit left from 𝐴 and two units down from 𝐴.
So now if we find the position of
𝐵 with respect to this origin, we see that the point 𝐵 is one unit length left
from the origin 𝑂, that is, negative one unit, and is on the horizontal or 𝑥-axis,
which means we have zero units in the 𝑦-direction. The coordinates of 𝐵 are therefore
negative one, zero.
We’ve seen how to define an
oblique, an orthogonal, or an orthonormal coordinate plane and how to define a point
in such a plane. Let’s complete this video by
recapping some of the key points covered.
𝐴 coordinate plane is formed by
any three noncollinear points 𝑂; 𝐼, 𝐽, where 𝑂 is the origin, the line 𝑂𝐼 is
the 𝑥-axis with positive direction 𝑂𝐼, and 𝑂𝐽 is the 𝑦-axis with positive
direction 𝑂𝐽. The lengths of line segments 𝑂𝐼
and 𝑂𝐽 are the unit lengths of the 𝑥- and 𝑦-axes, respectively. There are three main types of
coordinate planes. These are oblique, where the axes
are not perpendicular, orthogonal, where the axes are perpendicular but the unit
lengths need not be the same, and orthonormal, where the axes are perpendicular and
the unit lengths are equal.
The orthonormal plane is the
standard coordinate plane used in mathematics. The position of any point 𝑀 in
coordinate plane 𝑂; 𝐼, 𝐽 is specified by its coordinates 𝑥 sub 𝑀, 𝑦 sub 𝑀,
where 𝑥 sub 𝑀 is the real number on the 𝑥-axis at the point of intersection of
the line parallel to the 𝑦-axis and through 𝑀. And 𝑦 sub 𝑀 is the real number on
the 𝑦-axis at the point of intersection of the line parallel to the 𝑥-axis and
through 𝑀.
