Vectors, dot product, and vector components

Read LA 1.0-1.2, 4.2.0, supplementary notes, pp. 1 - 4.

1) Main points

This chapter introduces the concept of vectors. A vector is a line with a length and direction. They are marked as two numbers over each-other. 3 over 4 would mean a line from (0,0) to (3,4). They can be multiplied by constants to double the length of the line. They can also be added together, the top number plus the top number over the bottom plus the bottom. Graphically they can be added by using the parallelogram or head to tail method. This involves putting the tail of one on the head of the other and wherever it points is the sum. 4.2 introduces the operation of a dot product which is when you are multiplying two vectors together it becomes the sum of the top numbers plus the sum of the second numbers and so on. If the vectors are perpendicular then the dot product is zero. The supplementary notes summarize some of the information in 1.0-1.2. Because of the pythagorean theorem, the length of a vector is the square root of a squared + b squared. Using our existing knowledge of partial derivatives you can find the directional derivative.


2) Challenges

Dot products may present some challenges if I am not careful in writing them out or realizing the answer should be a scalar. I don't really understand the directional derivative, I see where it's coming from but I think I need a better description of it.

3) Reflections

I used vectors and scalars in physics a ton!! They were useful in determining whether an object had a direction or not and whether to pay attention to the sign of a variable. I can see vectors being useful in all sorts of real life applications.