SUNY Geneseo Department of Mathematics

Vectors

Thursday, September 8

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About?

Length of appointments? Although I’m not sure I said it explicitly (except maybe in our conversation about course policies), most of you seem to have decided that half an hour is a good length for a grading appointment. It turns out that’s exactly what I’d like too, so please make appointments for grading half an hour long. We might not always use the whole time, but it should be enough that we’ll have time for any questions you or I want to ask, etc.

Vectors

Based on “Working with Vectors in R³” in section 1.2 of the textbook.

Questions?

Finding unit vectors?

The basic theory behind finding unit vectors is that if you divide the components of a vector by that vector’s length (or, in more formal terms, scalar multiply the vector by the reciprocal of its length), you get a unit vector:

Dividing every element of a vector by its length and taking the length of the result gives 1

As an example, try the calculation on the vector v = ⟨4, 1, -4⟩.

Unit vector in direction 4, 1, negative 4 is vector 4 over root 33, 1 over root 33, negative 4 over root 33

Key Ideas

Vector arithmetic

Addition/subtraction
Scalar multiplication
Length

A unit vector in the direction of v is v scaled by 1/|v|.

Examples

Warm-Up

Let u = ⟨1, -2, 2⟩ and v = ⟨4, 1, -4⟩. Calculate…

u + v

Sum of vector 1, negative 2, 2 and vector 4, 1, negative 4 is vector 5, negative 1, negative 2

3 times vector 1, negative 2, 2 is vector 3, negative 6, 6

v - 4u

Vector 4, 1, negative 4 plus 4 times vector 1, negative 2, 2 is vector 0, 9, negative 12

Length of vector 1, negative 2, 2 is square root of 1 squared plus negative 2 squared plus 2 squared, or 3

Lengths and Addition

Vectors obey something called the “triangle inequality,” which says that if u and v are vectors, then |u+v| ≤ |u| + |v|.

Vectors V and U form 2 sides of triangle with other side is V plus U; sum of lengths is at least length of sum

Can you find one or more examples of vectors whose sum is shorter than the sums of the lengths of the individual vectors? The u and v vectors from the previous examples work well:

Length of vector 1, negative 2, 2 plus vector 4, 1, negative 4 is square root of 30 which is less than sum of lengths

How about examples of vectors whose sum has a length equal to the sums of the lengths of the individual vectors? The 0 vector is a simple example:

Length of vector V plus zero vector is length of V which equals length of V plus length of 0 vector

In general, the length of the sum equals the sum of the lengths when the vectors are parallel, i.e., point in either the same or exactly opposite directions.

Problem Set

Problem set 2, on vectors and a little bit of plotting related to quadric surfaces, is available.

Roughly speaking, the idea is to work on it next week, and grade it during the week after.

See the handout for details.

Imagine 2 vectors that point in more or less the same general direction, but that aren’t perfectly parallel. Is there some sense in which you can talk about the “part” of one of those vectors that points in exactly the same direction as the other?

Vectors pointing in roughly same direction, with projection of one into the other's direction

Monday we’ll look at an operation on vectors that’s helpful for answering this question (and others): the dot product.

Please read “The Dot Product and Its Properties,” “Using the Dot Product to Find the Angle between Two Vectors,” and “Projections” in section 1.3 of the textbook.

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