Show that two Lines are Skew

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Determine that the lines L1:

and L2:

are skew.

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: 0, –2, and 0.

State the coordinates of this point.

A has coordinates (0, –2, 0).

Add this point to your diagram. Then check your work by clicking "Next".

Note the use of color for the coordinates of the point.. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Now we need to plot a vector, , in the direction of the line L1.
We'll also 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 as components of the vector.

What are the values of those components here?

The values are 3, –3, and 6.

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. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

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: 5, –2, and 0.

State the coordinates of this point.

C has coordinates (5, –2, 0).

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. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Are the lines skew?

No, they appear to intersect.

Is this visual check sufficient?

No.

What else can we do?

We can check that the lines have no common point.

Let's assume that the x and y-values are equal. How can we use that idea?

From the equations of thelines 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 are skew, the z values will be different, indicating that the lines do not have a common point and hence, do not 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. Solve for t in 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

Find t by substituting Equation 5 into Equation 3.

We get

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 value of the parameters which produce different z values for the lines. Hence the lines do not intersect.

Can we conclude that the lines are skew?

No, the lines might be parallel.

How do we determine that?

If the lines are parallel, vectors along them must be proportional.
Consequently, we need to use the above vectors
and
.
Are they proportional?

No.

What is our conclusion?.

The lines are neither intersecting nor parallel. They are skew.

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

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