Angle between 2 Planes

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Find the acute angle between the two planes defined by

How do we proceed?

We find the normal vectors, , to the planes and then find the acute angle between them.

How do we find the angle?

We use the dot product of the normals and their lengths, .

State this method.

First we find its cosine:

Then what?

We use the inverse cosine to find the angle.

Let's graph this data on paper. To plot plane P1, 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.

Find the y-intercept in a similar way.

Find the z-intercept in a similar way.

Show these points on a graph and connect them to show plane P1. Draw the x-axis out of the plane of the paper, as usual. Then check your graph by clicking on "Next".

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Determine the intercepts for plane P2.

We get

Show these points on a new graph and connect them to show plane P2. Then check your graph by clicking on "Next".

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Next, let's graph the normals separately. Again we'll use an x–axis out of the plane of the diagram and place the base of the normal at the origin.
How do we get ?

We need the direction numbers of plane P1.

What are the direction numbers in general?

The direction numbers of a plane are the coefficients of x, y, z in the standard equation of the plane.

What are their values for P1?

For P1, the direction numbers are

Graph the normal vector, using these values as the lengths of the components of it.

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Note the use of color for the three components.

Use a similar analysis to state the values of the direction numbers for P2.

For P2, we find

Graph , using these values as the lengths of the components of it.

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Set up the dot product of two vectors using their general components.

Evaluate the dot product using the values found above.

We also need the length of each vector. Set up the calculation of the length of .

Substitute and evaluate.

Find the length of

Combine these results to find the cosine of the acute angle between the normals.

Determine the angle.

How is this angle related to the acute angle between the planes?

It is the same.

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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