Volume of a Parallelepiped via the Scalar Triple Product:

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Find the volume of the parallelepiped with adjacent edges AB, AC, and AD, where the points are A(6, –3, –6), B(6, –3, 0), C(3, –3, –2), and D(–1, 8, –4).

Let's graph this situation. First we'll draw a 3-Dimensional set of axes, with the x axis out of the plane of the paper. Sketch the axes on paper. Then check your work by clicking "Next".

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Now add Point A. Then check your work by clicking "Next".

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Now add Point B. Then check your work by clicking "Next".

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Now add Point C. Then check your work by clicking "Next".

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Now add Point D. Then check your work by clicking "Next".

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Note that line AB is vertical. We can consider points A, C, and D as defining 3 vertices of the bottom of the parallelepiped. Show the bottom by drawing the lines AC and AD, and lines parallel to them. Then check your work by clicking "Next".

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Add vertical edges by drawing AB and similar lines at the other vertices of the bottom. Then check your work by clicking "Next".

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Now construct the top of the parallelepiped by connecting the tops of the verticals. Then check your work by clicking "Next".

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Finally, show the vectors AB, AC, AD. Then check your work by clicking "Next".

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How do we proceed to get the volume of the parallelepiped?

We determine the magnitude of the Triple Scalar Product.

State that in terms of AB, AC, AD.

One way to express this is

So we need to obtain these vectors. How do we do that?

We use the difference of the x-coordinates of points B and A to find the x-component of vector AB, etc.

Find the vector AB using A(6, –3, –6) and B(6, –3, 0).

We get

Find the vector AC using A(6, –3, –6) and C(3, –3, –2).

We get

Find the vector AD using A(6, –3, –6) and D(–1, 8, –4).

We get

Next, set up the determinant for the cross product in terms of the symbolic components .

Substitute values.

Convert this to factors.

We get

Simplify.

Now set up the dot product

Simplify.

We get Volume = 198.

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