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Show that two Lines Intersect
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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:
intersect.
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: 7, –2, and 5.
State the coordinates of this point.
A has coordinates (7, –2, 5).
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,
, in the direction of the line L1. We'll extend the vector to show the line. What part of the parametric equations do we use for this?
We use the coefficients of the parameter "s" as components of the vector.
What are the values of those components here?
The values are –3, –1, and 0.
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. Using its parametric equations, how do we determine the coordinates of the reference point, C, for line L2?
We use the three constants: –1, 1, and 2.
State the coordinates of this point.
C has coordinates (–1, 1, 2).
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 "t".
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 intersect?
They appear to intersect.
Is this visual check sufficient?
No.
What else can we do?
We can check that the lines have a common point.
Let's assume that the x and y-values are equal. How can we use that idea?
From the equations of the lines we can solve for the required values of the parameters
s
and
t
.
What is the next step?
We use those values of the parameters to calculate the corresponding value of
z
for each line.
If the lines intersect, the
z
values will be the same, indicating that the lines do have a common point and hence, do intersect.
Let's try this. Set up the equality for the x coordinates.
Equation 1
Set up the equality for the y coordinate.
Equation 2
We have two equations in two unknowns. Subtract 3 times Equation 2 from Equation 1.
We get
Equation 3
Use this result to replace
t
in Equation 2.
Equations 2 and 3 combined give us
Equation 4
Solve for
s
.
We get
Equation 5
Now what?
We can substitute these values of the parameters into the equations for
z
.
Do that for L1.
For L1:
Proceed similarly for L2.
We get
What is our conclusion?
Forcing x and y to be equal requires values of the parameters, which then produce the same z value for the lines. Hence the lines do have a common point and do intersect.
Let's check this graphically by plotting the point of intersection, I.
We have just determined its z-coordinate. How do we find its x-coordinate?
We substitute the value of either parameter into equation 1.
Do that. using
s
= 2.
We get
as the x-coordinate of the point of intersection.
How do we find its y-coordinate?
We substitute the value of either parameter into equation 2.
Do that. using
t
= –1.
We get
as the y-coordinate of the point of intersection.
State the coordinates of the point of intersection.
The point of intersection is (1, –4, 5)
Add this point to your paper graph. Then click "Next" to check.
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Do we get confirmation?
Yes.
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