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Two Vectors, not Orthogonal nor Parallel in 3D:
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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.
Show that the vectors
, and
are neither orthogonal nor 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 different from the values for 0 and 90 degrees.
What are those values for these angles?
The values are 1 and 0, respectively.
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".
Note the use of color for the coordinates of the head of the vector.
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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 or perpendicular?
Since this value is neither 0.0 nor 1.0, we can conclude that the vectors are neither parallel nor perpendicular.
Just for interest's sake, determine the angle between the vectors.
What does this tell us about the two vectors?
Since the angle is neither 0 nor 90 degrees, this also tells us that the vectors are neither parallel nor orthogonal.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index