Line through Two Points:

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Find the vector, parametric, and symmetric equations for the line L through A(7,–2, 5) and B(–5,–6,5).

Let's plot this on paper, using the usual 3-D axes with the x-axis out of the plane of the diagram. To begin, plot the point A. Then check your work by clicking "Next".

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Now plot the point B. Then check your work by clicking "Next".

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Add the line AB to your diagram. Then click "Next" to check.

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How do we get a vector equation for the line?

One way is to construct
1) a vector, , from the origin to point B, and
2) a vector, , along the line.
3) Then we add a multiple, t of the second vector to the first vector to get a vector to any point on the line:

How do we get a vector in the direction of the line?

We use the coordinates of the two points to obtain the vector .

How do we use this to express any point on the line?

We use a multiple of it, .
Any real number can be used for t.

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

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Add to your diagram. Then click "Next" to check.

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Now show the vector sum

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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 t, which need not be an integer.

What are the components of .

They are the same as the coordinates of point B.

Express as a vector.

Using the coordinates of B, we get

How do we determine the components of the vector ?

We use the difference of the coordinates of points A and B.

Determine the x-component of .

The x-component is

Determine the y-component of .

The y-component is

Determine the z-component of .

The z-component is

Express in terms of these components.

Combine these results to express the vector equation of the line.

We add the two vectors and simplify:

Use this result to get the parametric equation of the x-coordinate of the line.

Let (x,y,z) be the coordinates of a point on the line. Let t be the parameter. Then

Get the parametric equation of the y-coordinate of the line.

Get the parametric equation of the z-coordinate of the line.

Let's check the parametric equations by calculating the coordinates of the point associated with . Then we can plot that point to compare it with the graph of the line.
Find the coordinates of this point.

We get

Plot this point (1,–4, 5) on your graph of the line. Then check by clicking "Next".

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Does that point appear to be on the line?

Yes.

Is this a guarantee?

No.

Why?

The point may be in front of or behind the line in this view.

How can we check with a different graph?

We could rotate the axes so that we look at the graph from what is now the right-hand side. Then if the point is not on the line, we will see that.

How should we draw the new graph?

We can have the positive x-axis point to the left.

If we keep the positive z-axis upward and rotate about it, what will be the new direction of the positive y-axis?:

The positive y-axis will then be out of the plane of the diagram, shown in our usual slanted way.

Draw these new axes on paper. Then click "Next" to check your work.

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Now show A, B, and the line.

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Use the coordinates (1,–4, 5) to show point C.

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What can we conclude?

The point C shows to be on the line in both views. We must have the correct coordinates for it.

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