Line through a Point parallel to a Line:

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Find the parametric, symmetric, and vector equations for the line L1 through A(9, –2, 3) and parallel to the line L2 having the parametric equations
.

How do we proceed?

To find L1, we need a vector to combine parametrically with the coordinates of point A.

How can we get such a vector?

Since L1 must be parallel to L2, we can use a vector which describes L2.

How can we get that vector from the parametric equations for L2?

We can use the fact that the coefficients of the parameter "t" in the parametric equation are the components of a vector along line L2.

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 plot the point A. 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 let's show a point, C, on line L2. Using the parametric equations, how do we determine the coordinates of this reference point?

We use the three constants: 3, 5, 1.

State the coordinates of this point.

C has coordinates (3, 5, 1).

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.

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 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 V 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!

Can we use this vector to obtain line L1?

Yes.

How?

We can duplicate it based at point A. Add it to your diagram and check your work by clicking "Next".

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How do we get the constants for the parametric equation for line L1?

We use the coordinates of point A as the constants.

How do we get the coefficients of the parameter for L1?

We use the components of the vector V.

Should we use a different symbol, instead of "t", for the parameter for L1?

Yes, since we are describing a different line.

Combine these concepts to express the parametric equations for line L1.

parametric equations

How do we find the symmetric equations for the L1?

We solve the parametric equations for s.

Do that.

We get

symmetric equations

How do we get a vector equation for the line?

One way is to use
1) a vector, , from the origin to point A, and
2) a vector along the line, such as V..
Then we add the two vectors.

Show in your diagram. Then click "Next" to check.

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Now show the vector sum with s = 1.

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Summarize what R represents.

R describes any point on the line. The various locations of points on the line L are associated with corresponding values of the parameter s.

Relate the general vector sum to the data given in the problem.

We get the vector equation for the line L1:

Collect the coefficients of i, j, k on the right-hand side.

vector equation

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