SUNY Geneseo Department of Mathematics

Introduction to Vectors

Monday, February 6

Math 223
Spring 2023
Prof. Doug Baldwin

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Anything You Want to Talk About?

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Meetings

Don’t forget to make an appointment (and thank you to those who already have).

My office is South 307, which is in the “nose” part of South Hall that sticks out towards the street:

Map of 'L' shaped hall with diagonal bit sticking off corner; X marked near lower left end of diagonal

3-Dimensional Vectors

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

Key Ideas

The rules/formulas for vector addition, subtraction, and scalar multiplication. All add, subtract, or scale vectors component by component.

Unit vectors are vectors whose magnitude is 1; you can find a unit vector in the direction of a non-unit vector by dividing each component of that vector by the vector’s magnitude.

A vector’s “magnitude” is its length, and is calculated via a formula based on the distance formula. If v is a vector, the notation ||v|| means the magnitude of v:

Magnitude of vector X Y Z is square root of X squared plus Y squared plus Z squared

Questions

What are the standard unit vectors i, j, and k and how do they compare to component notation?

The standard unit vectors are vectors of length 1 that point along the axes. Any vector can be calculated by multiplying these vectors by appropriate scalars and adding the products. In fact, the scalars to multiple by are exactly the x, y, and z components of the vector:

3 dimensional coordinate axes with short arrow labeled I hat on X axis, J hat on Y, and K hat on Z

Magnitude?

Here’s an example of calculating magnitude, connected to the distance formula by the idea that any vector ⟨x, y, z⟩can be thought of as extending from the origin to point (x, y, z):

3 dimensional coordinate axes with vector 5 9 negative 4 going from origin to point 5 9 negative 4

Vector operations?

Here are some examples of calculations on vectors. Note that they all extend similar rules for 2 dimensions in a natural way:

Vector addition adds matching components of 2 vectors; scalar multiplication multiplies each component by scalar

Finding a unit vector in the direction of some other vector, v?

The book gives a formula, namely 1 over the magnitude of v times v. Let’s see that this formula works, in the sense that it always produces a vector that’s a scalar multiple of v and therefore parallel to it, and that has length 1:

Magnitude of vector X Y Z times 1 over its magnitude is 1

Here’s an example of applying the formula:

Unit vector in direction 1 1 4 is 1 over 3 root 2 times 1 1 4 or 1 over 3 root 2, 1 over 3 root 2, 4 over 3 root 2

Next

An example of vector calculations similar to the football one from the book.

Then an introduction to the dot product, an operation on vectors that’s helpful for solving problems such finding angles between vectors, breaking vectors into sums of other vectors with desired directions, etc.

Please read “The Dot Product and Its Properties” in section 1.3 of the textbook.

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