SUNY Geneseo Department of Mathematics

The Principal Unit Normal

Wednesday, October 3

Math 223 01
Fall 2018
Prof. Doug Baldwin

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Previous Lecture

Misc

Actuary Interest Meeting

Today, 3:00 - 4:00, South 328

Learn what an actuary is and meet the actuary club

First Hour Exam

Friday, October 5, in class.

Covers material since the beginning of the semester (e.g., 3D space; lines, planes, and other surfaces; rectangular, polar, and cylindrical coordinates; vectors and their operations; vector valued functions and their calculus). Arc length and curvature may appear, but will probably be to some extent deferred to a future exam.

3 to 5 short-answer questions.

You’ll have the full 50-minute class period.

Open book, notes, computer as a reference or calculator; closed person.

Sample exam and solution are now available.

Questions?

The Principal Unit Normal

Calculation

Find the principal unit normal to r(t) = 〈 -sin t, cos t, √3 t 〉.

The principal unit normal is the derivative of the unit tangent vector, scaled to unit length. So the basic calculation involves computing and scaling 2 derivatives.

Formulas for unit tangent and principal unit normal

Calculate the unit tangent vector by differentiating and scaling r:

Unit tangent is derivative of of r over magnitude of derivative

Then calculate the principal unit normal by differentiating and scaling T:

Principal unit normal is derivative of tangent over magnitude of derivative

Physical Significance

What’s the geometric relationship between a constant-magnitude vector and its derivative? Figure it out by starting with an expression for magnitude (or at least its square), then take derivatives:

Derivative of dot product, via product rule, is 0 means r dot r-prime is 0

They’re perpendicular!

So the principal unit normal is always perpendicular to the tangent vector. Furthermore, because the principal unit normal is a derivative, it points in the direction the tangent is changing towards, i.e., the direction the curve is turning.

Curve with tangent vectors and perpendicular normal vectors

This particularly has implications for velocity and acceleration vectors: You can decompose acceleration into a component parallel to the tangent, which is the portion of acceleration responsible solely for speeding up or slowing down the object (since the tangent vector is parallel to velocity), and a component parallel to the principal unit normal, which is solely responsible for changing the object’s direction.

Position with velocity and acceleration, in parallel and perpendicular parts

Key Points

Principle unit normal as a vector that gives direction towards which a curve turns.

Formula for the principal unit normal.

Next

Multivariable functions.

Read section 4.1.

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