Plane through 3 Points: Example 3
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Example 1
Example 2
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Find the plane through the three given points:
A
(–2, –2, 3),
B
(2, –1, 2), and
C
(3, 6, 2).
Let’s graph this data on paper. Draw the x-axis out of the plane of the paper, as usual.
First plot point
A
. Then check your graph by clicking on “Next”.
In this diagram, note the colors used to show the three coordinates.
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Plot point
B
. Then check your graph by clicking on “Next”.
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Plot point
C
. Then check your graph by clicking on “Next”.
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Connect these points to show the plane. Then check your graph by clicking on “Next”.
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We need the normal vector, n, in order to describe the plane. How can we obtain it?
If we have two vectors in the plane we can form their cross product. That result is perpendicular to each vector, and hence perpendicular to the plane.
How can we get two vectors in the plane?
We can use, for example, the vector from
A
to
B
and the vector from
A
to
C
.
Call these
.
Express the
x
-component,
AB
x
, in terms of the symbols for the coordinates of the points
A
and
B:
A
x
, A
y
, A
z
, B
x
, B
y
, and
B
z
.
Evaluate this using the data from the problem.
We get
In a similar way, evaluate
AB
y
.
In a similar way, evaluate
AB
z
.
Use this information to write
AB
in terms
of the unit vectors
i
,
j
, and
k
.
We get
AB
= 4
i
+
j
–
k
In a similar way, determine the three components of
AC
.
Use this information to write
AC
in terms of the unit vectors
i
,
j
, and
k
.
We get
AC
=
5i
+ 8
j
–
k
Now we are ready to find the normal vector by forming the cross product of these two vectors. Set up
in determinant form.
Evaluate the determinant.
We get
Add n at point
C
in your diagram. Then click “Next” to check your work.
Note that the vector is roughly twice as long as the part which is shown.
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How can we use this normal vector to get an equation for the plane?
The components of the normal vector are also the direction numbers
a, b, c
.
Express this in an equation.
Use this information in the linear equation of the plane.
How do we determine d?
We can use the coordinates of any point in the plane.
Substitute the coordinates of point
A
.
As a check, use the coordinates of
B
in the linear equation of the plane.
We get
Finally, check our work by using point
C
.
Next, Let’s determine the scalar equation of this plane.
State that equation in general, using the general point
.
The scalar equation of the plane with the normal
is
Do the
a, b, c
have the values from above?
Yes,
How do we find
?
We can use the coordinates of any point in the plane.
Substitute the coordinates of point
C
.
We get
Check this by substituting the coordinates of point
B
as the values of
x, y, z
We get
, as expected.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index