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Show that two Lines are Parallel
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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.
Determine that the lines
L1:
and
L2:
are parallel..
How do we proceed?
If the lines are parallel, vectors along them must be proportional.
Consequently, we need vectors parallel to each line.
How can we get such a vector,
, from the parametric equations for L1?
We can use the fact that the coefficients of the parameter "t" in the parametric equation are the components of a vector along line L1.
How can we find the vector,
, from the symmetric equations for line L2?
We can convert the symmetric equations to parametric equations and use the method just described.
First, 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 let's show a point, A, on line L1. Using the parametric equations, how do we determine the coordinates of this reference point?
We use the three constants in the parametric equations: 3, 5, –4.
State the coordinates of this point.
Point A has coordinates (3, 5, –4).
Add this point to your diagram. Then check your work by clicking "Next".
Note the use of color for the coordinates of the point..
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Now we need to plot a vector,
, parallel to line L1. What part of the parametric equations do we use for this?
We use the coefficients of the parameter as components of the vector.
What are the values of those components here?
The values are –2, 1, and 3.
State this vector in component form.
Should the vector be draw from the origin?
It can be drawn anywhere. Let's have its base at point A.
Add the vector to your diagram. Check by clicking "Next"
Note that the vector is extended in light blue in both directions to indicate line L1. The components of
are also shown.
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Now let's show a point, C, on line L2. To get its coordinates we need to convert the given symmetric equations to parametric equations. How do we do that?
We set a parameter equal to the symmetric equations and solve for an expression for each of the variables.
Should we use a different parameter,
s
?
Yes.
Why?
Line L2 and its equations are independent of any conditions on line L1, so an independent parameter is needed.
Set up the new parameter, equal to the symmetric equations.
Solve for
x
.
Solve for
y
.
Solve for
z
.
Using these parametric equations, how do we determine the coordinates of the reference point, C, for line L2?
We use the three constants: 1, –1, 8.
State the coordinates of this point.
Point C has coordinates (1, –1, 8).
Add this point to your diagram. Then check your work by clicking "Next".
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What is the direction of line L2?
It is in the direction of a vector with components obtained from the parametric equations for line L2.
What part of the parametric equations describes this vector?
The vector is described by the coefficients of the parameter "s".
State this vector in component form.
Should this be drawn with its tail at the origin?
No, it should be drawn with its tail based at point C.
Add this vector to your diagram. Then click "Next" to check your work..
Note that the vector is extended in light blue in both directions to indicate line L2. The components of
are also shown.
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Do the lines appear to be parallel?
Yes.
Is this visual check sufficient?
No.
What else can we do?
We can compare the equations for the two vectors.
What do we look for?
If the lines are truly parallel, one equation will be proportional to the other.
Compare the two equations.
The vectors are proportional, so the vectors are parallel.
What about the lines?
L1 and L2 are also parallel.
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