Plane through a Point Containing a Given Line
General Contents
Detailed Contents
Index
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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 point
containing the line L:
Lets visualize this data by graphing it on paper. Show the
x
-axis as out of the paper as usual. Plot point
A
. Then check by clicking Next.
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Calculate points on the line at
t
= -1,
t
= 0,
t
= 1,
t
= 2.
We get
Plot these points and add the line to your diagram. Then click Next to check.
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How do we proceed?
To get the equation of the plane, we need its normal.
How can we get that?
We can get it from the cross product of two vectors in the plane.
How can we get such vectors?
We could use point
P
and two other points in the plane to set up the two vectors.
Where can we get two points?
Since the line is in the plane, we can use points
B
and
C
on the line.
To emphasize this, show the plane by connecting points
P
,
B
, and
C
. Then click Next to check.
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Now we can set up our vectors. Set up PB in terms of symbols for the coordinates of the points.
Substitute the values of the coordinates and determine the numerical values of the components of PB.
Determine the numerical components of PC in a similar way.
Now we can form the cross product of these two vectors. What will be the direction of the cross product, relative to the plane?
The cross product is perpendicular to the plane of the two vectors. Since the two vectors are in the desired plane, the cross product will be perpendicular to them and hence to the plane. We can then use a multiple of it as the normal.
Set up the cross product in determinant form.
Determine the components.
For graphing purposes, lets use an approximate multiple of this vector as the normal. Dividing by 4, we get approximately
Add this vector to your diagram, with the base of the vector at point
P
. Then click Next to check.
Note that this is an approximation to the direction of the real normal.
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How do we use the normal to get our equation of the plane?
The components of the true normal,
,
are the direction numbers
a, b, c
.
State the general equation of the plane in terms of
Substitute the numerical values.
Using the actual normal,
we get
Simplify.
The equation of the plane is
Should this equation apply if we substitute the coordinates of some other point in the plane?
Yes.
Check this equation by using the coordinates of point
P
.
It checks!
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index