SUNY Geneseo Department of Mathematics

Lines in Space

Monday, February 13

Math 223
Spring 2023
Prof. Doug Baldwin

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Anything You Want to Talk About?

(No.)

Colloquium

There’s a math colloquium this week, and I’m the speaker.

The talk is an introduction to the math behind a certain kind of computer graphics, hoping to show that graphics pulls together a surprising amount of a typical undergraduate math curriculum.

Thursday, February 16, 3:00 - 4:00 PM, Newton 201.

Lines

Based on “Equations for a Line in Space” and “Distance between a Point and a Line” in section 1.5 of the textbook.

Key Ideas

Lines can be described by a vector equation, parametric equations, or symmetric equations (which are all basically alternative forms of the same thing). See the questions below about these equations for examples.

There’s an equation for finding the distance between a point and a line. To be explained tomorrow.

In parallel vectors, corresponding components all have the same ratio.

Questions

The vector, parametric, and symmetric equations, and examples?

Start with the idea that if you take a point, and add various multiples of a vector to it, you get new points lying on a straight line through the original point, and in the direction of the vector (or its negative, if the multiples are negative):

Point P with vector V through it define a line of points

The idea that any multiple of the vector produces a point on the line leads to the vector equation for the line:

Vector R of T equals P plus T times vector V

Writing the vector equation out component by component and then separating the equations for each component gives the parametric equations for the line:

X of T equals X sub 0 plus T X sub V, Y of T equals Y sub 0 plus T Y sub V, Z of T equals Z sub 0 plus T Z sub V

And solving each parametric equation for t gives a set of 3 expressions that must all be equal to each other, since they all equal t. These are the symmetric equations for the line:

$X of T minus X sub 0 over X sub v equals T equals similar fractions involving Y and Z$

For example, give the equations for the line through point (1, 0, 3) in direction ⟨-2, 6, 1⟩.

Once you identify how the components of the point and vector match up with the variables used in the equations, you can read the equations out:

Point and direction vector with vector, parametric, and symmetric equations for a line built from their components

On the last line here, notice that you can negate the numerator and denominator of one of the symmetric expressions to get rid of a negative denominator if you want.

Also notice that the equation for a line isn’t unique. For example, I can double the direction vector and and say that the line above has vector form r(t) = (1, 0, 3) + t ⟨-4, 12, 2⟩. This is a correct equation, although each point on the line will correspond to a different t value than in the original equation.

As another example, what’s an equation for the line through points (2, -2, 3) and (5, -4, 2)?

Subtract one point from the other to get a direction vector, then read parametric (or any other form you like) equations from it and one of the points. You’ll get different equations, but all for the same line, if you subtract points in the opposite order or use the other point as the reference.

Line and vector through 2 points. X equals 2 plus 3 T, Y equals negative 2 minus 2 T, Z equals 3 minus T

More about lines, in particular the distance formula, and maybe a bit about determining whether lines intersect.

No new reading, but review the material on lines if you want.

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