Plane Parallel to Another Plane through a Given Point
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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 , P1, through A(3,1,2) parallel to the plane P2:
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 and later diagrams, colors will be used to show the three coordinates: blue for
x
, yellow for
y
, and green for
z
.
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To plot the plane, we can use its intercepts on the axes. How can we find its x-intercept?
Since
for all points on the x-axis, we can substitute those zero values into the equation of the plane and solve for
.
Do that.
Show this point on a new graph. Then check your graph by clicking on “Next”.
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Find the y-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on “Next”.
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Find the z-intercept in a similar way.
Add this point to your diagram. Then check your graph by clicking on “Next”.
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Connect these three points to show the lines of intersection of the given plane with the coordinate planes. Then check your graph by clicking on “Next”.
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We defined
a, b,
and
c
above. What name do we give to this group of properties of a plane?
They are the direction numbers of the plane.
We are looking for a new plane parallel to the given plane. What do we know about its direction numbers?
They are the same as those of the plane it is parallel to.
What form of the equation of the plane does this suggest?
The scalar form.
We need a reference point for that equation. What point can we use here?
Since the reference point must be in the plane, we can use point A, which the problem requires to be in the new plane.
Set up the scalar equation of the new plane using the symbols
We get
Substitute the numeric values.
The equation becomes
Rewrite this in the standard form.
We get
So now we have the equation of the desired plane through point A. Determine its intercepts as we did above.
On the
x
-axis,
y
and
z
are 0, and we have
On the
y
-axis,
x
and
z
are 0, and we have
On the
z
-axis,
x
and
y
are 0, and we have
Plot these on a new graph and connect them to show our desired plane. Then check your graph by clicking “Next”.
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Let’s add the normal to the plane in the diagram. What are the components of the normal?
The direction numbers,
a, b,
and
c
can be used for the components.
Add the normal with its base at (0,2,4) and point A to the previous diagram. As usual, check your diagram by clicking “Next”.
Note that point A is in the plane of the triangle, and that the normal is perpendicular to the plane.
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As another way to check our work, use the intercepts for both planes to show a triangle for each of them. Do this on a new diagram. Then check your work by clicking “Next”.
As mentioned before, the edges of the triangles represent the intersection of the planes with the coordinate planes. They are parallel, as required.
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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