Linear Algebra 4.1 - 4.2

Main Points:
Dot products can be used to find the lengths of vectors and the angle between them. The length of a vector can be calculated by taking the square root of the sum of the squares of the components of the vector. The cosine of the angle between two vectors is given by dividing the dot product of the two by the product of their lengths. When a vector isn't a multiple of another, one can figure out the closest the vector can get to the intended point. To do this, one must utilize two equations. The first is that the closest the vector can get (which is represented by another vector r) is the point perpendicular to the intended point, so r dotted with the vector trying to be reached equals zero. The second equation is formed from the fact that a multiple times the vector plus the vector r between the two must equal the vector trying to be reached.

Challenges
The biggest hang up of this reading for me was the section describing how to find if two vectors are perpendicular. I understand that the product of the slopes of the vectors equals negative one when the vectors are perpendicular, but the way the reading went about showing this was confusing.

Reflections
This concept of finding an answer that is the best possible, rather than flat out the best brings about some useful applications. We've been finding optimized solutions for problems given restraints, and this seems like it could be used in a similar way.