Plane through a Point with a given Normal

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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 the point

with normal vector the normal

Let’s graph this data on paper. Show the x-axis out of the plane of the diagram as usual. Plot point A. Then check your work by clicking on “Next”.

Note the use of blue, yellow and green for the coordinates of point A. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Next let’s add the normal vector with its base at point A. Check your diagram by clicking “Next”.

Here the colors show the components of the normal. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

How do we proceed to get the equation of the plane?

We can use the scalar equation of the plane.

Using the symbols

, state the scalar equation of the plane.

general scalar equation of a plane in 3D

Substitute the values of the symbols.

We get

Simplify.

We get

How can we check this result?

Since this applies to all points in the plane, it must apply to point A. We can substitute its coordinates and expect to get 24.
Do that.

It checks.

For a visual check, let’s plot the intercepts of this plane with the axes and then connect them to show the orientation of the plane. 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 graph. Draw the x-axis out of the plane of the paper, as usual. Then check your graph by clicking on “Next”.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Find the y-intercept in a similar way.

Add this point to your diagram. Then check your graph by clicking on “Next”.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Find the z-intercept in a similar way.

Add this point to your diagram. Then check your graph by clicking on “Next”.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Connect these three points to show the lines of intersection of the given plane with the coordinate planes.

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We see from this graph that the plane seems to be oriented correctly relative to the normal and contains point A.

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

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