General Contents
Detailed Contents
Index
Show that two lines are perpendicular:
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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 line L1 through the points A(7, –2, 3) and B(–5, –6, 5) is perpendicular to the line L2 through C(–1, 1, 2) and D(–3, 6, 0).
How do we proceed?
We need to show that vectors along each line are perpendicular.
How do we get such vectors?
We use the difference of the coordinates of A and B to get
AB
, and
the difference of the coordinates of C and D to get
CD
.
How will we know if the vectors are perpendicular?
If two vectors are perpendicular, their dot product is zero.
Why is that?
The dot product is proportional to the cosine of the angle between them. If that angle is a right angle, the cosine, and the dot product, are zero.
Let's draw a diagram. Plot point A on paper. Use the usual 3-dimensional axes with the x-axis slanting out of the plane of the diagram. Then check your graph by clicking "Next".
Note the coordinates shown in color.
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Add point B to your diagram, and then check by clicking "Next".
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Add point C to your diagram, and then check by clicking "Next".
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Add point D to your diagram, and then check by clicking "Next".
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Add the lines to the diagram, and then check by clicking "Next".
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Determine the x-component of
AB
.
Determine the y-component of
AB
.
Determine the z-component of
AB
.
Use this information to write
AB
in terms of components.
Determine the x-component of
CD
.
Determine the y-component of
CD
.
Determine the z-component of
CD
.
Use this information to write
CD
in terms of components.
Now set up the dot product in terms of the general components
.
Substitute the values we have just found.
Are the vectors perpendicular?
Yes, since their dot product is zero.
What is our conclusion?
The lines are also perpendicular.
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