Parallel Vectors in 3D

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Show that the vectors , and are parallel.

How do we proceed?

We can get the cosine of the angle between the vectors from the dot product and see if it is consistent with 0 or 180 degrees.

What are the values for the cosines of 0 and 180 degrees?

Cosine of 0 degrees is 1 and the cosine of 180 degrees is –1.

How do we get the cosine from the dot product?

We can set up the dot product of the vectors in two ways and solve for the cosine of the angle.

State those two ways in terms of the components and the lengths .

Solve for .

Can we express in terms of their components, too?

Yes.

State those expressions.

and

How do we eventually obtain the angle?

We use the inverse cosine to find the angle.

Let's plot this on paper, using the usual 3-D axes with the x-axis out of the plane of the diagram. Draw the axes and then click "Next" to check your work.

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Now show the vector A, based at the origin. Then check your work by clicking "Next".

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Now show the vector B, based at the origin. Then check your work by clicking "Next".

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Let's discuss the calculations summarized above in more detail. How do we find the Dot Product?

We find the Dot Product from the sum of the products of the components.

Set that up in terms of the general components of the vectors.

.

What are the values of the components of A?

What are the values of the components of B?

Substitute the numerical values.

Simplify this.

We get .

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

Substitute and evaluate.

Set up the calculation of the length of .

Substitute and evaluate.

Combine these results to find the cosine of the angle between the vectors.

What does this tell us about whether the vectors are parallel?

Since the cosine of 180 degrees is –1, the vectors are parallel, but in opposite directions.

Do the vectors look parallel or opposite in the graph?

They are opposite.

Let's rotate the graph so that we look in from the right. We can accomplish this by having the x-axis point to the left, and having the y-axis be perpendicular to the plane of the diagram.
Start a new diagram on paper with the axes so oriented. Then click "Next" to check.

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Add vector A to your diagram. Then click "Next" to check.

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Finally, add vector B to your diagram. Then click "Next" to check.

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Now do the vectors also look opposite?

Yes.

Can we be sure?

Determining the cosine of the angle between them is the surest test.

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

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