Lines

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Course Management

The grading deadline for problem set 1 is passed; I put a few temporary 0 grades for it into Canvas last night.

Any thoughts on how to “invite” contributions to class discussion? For example, the random names list, picking someone to call on, other...?

Lines in 3 Dimensions

Based on “Equations for a Line in Space” and “Distance Between a Point and a Line” in section 1.5.

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Key Ideas

Equations for the line through point P(x₀,y₀,z₀) in direction v = ⟨v_x, v_y, v_z⟩:

• Vector: r(t) = P + tv
• Parametric: x(t) = x₀ + tv_x; y(t) = y₀ + tv_y; z(t) = z₀ + tv_z
• Symmetric: (x-x₀)/v_x = (y-y₀)/v_y = (z-z₀)/v_z

Equations

Give an equation for the line passing through point (0, 3, 1) with direction vector ⟨1, 2, 3⟩.

Plug the point and vector directly into the vector form of the line’s equation:

Give an example of some point other than (0,3,1) that’s on that line. Each possible value of t gives a different point on the line, so just plug in some t values:

How about an equation for the line passing through points (0, 3, 1) and (1, 4, -1)?

There are several ways to come up with this equation. One is to use the changes in x, y, and z between the points in the parametric form of the line’s equation:

Another is to realize that those differences in x, y, and z are really the components of a vector from P₀ to P₁, and use that in the vector form of the line’s equation:

Once you have this vector, you can use it with either point, or you could use the negative of this vector (i.e., the vector from P₁ to P₀) with either point, etc.

And a very cool way to find the equation is to realize that the standard parametric and vector forms (which are really the same thing, the parametric form just separates the vectors out into their components) can be rearranged into a form that lets you read the equation straight from 2 points with no calculations:

Applications

Do the lines r₁(t) = ⟨4, -1, 3⟩ + t⟨-4, 2, -4⟩ and r₂(t) = ⟨0, 0, 4⟩ + t⟨2, 0, -3⟩ intersect? If so, where?

The basic idea here is to ask whether there is some point ⟨x,y,z⟩ (writing the point as a vector to it from the origin, since everything else here is a vector) such that r₁(t) = ⟨x,y,z⟩ for some t, and r₂(t) = ⟨x,y,z⟩ for some possibly different t. Since the t values are generally different between the two vector equations, we’ll try to find ⟨x,y,z⟩ by renaming the parameter for r₂ to something else (e.g., s) and then using the parametric form of the lines’ equations to require r₁ to equal r₂ and solving for t and s:

Notice that we had 3 equations and only 2 unknowns. We used 2 of the equations to find s and t, but we still need to check that the values we found really do work in the third. It’s possible to have a solution to 2 of the equations that doesn’t work in the third if the lines are skew. In this case we see that our values for s and t do also solve the last equation. Now you can use either s or t, in its corresponding parametric equations, to find the intersection point.

Next

If we understand lines in space, we should also understand the other “straight” thing in space: planes.

Please read “Equations for a Plane” in section 1.5 of the textbook.

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