Vector projection

Read LA 4.1 - 4.2.

1) Main Points

4.1 continues on our understanding of finding the solution to a linear combination by finding the solution of a vector equation. This chapter acknowledges that sometimes when solving a system of equations there is not a solution, so it introduces the concept of an approximate solution. The approximate solution is best understood by using vectors and the dot product. The dot product(review) looks like this:

(2,3) dot (4,2) = (2*4) + (3*4) = 8 + 12 = 20

The most important thing to remember about the dot product is that:

If the dot product of two vector is 0 then the vectors are Perpendicular (Orthogonal)

Because of this dot product we can use it and the Pythagorean theorem to learn a lot about the relationships of vectors. The length of vector u is equal to the square root of (u dot u), this is derived from the Pythagorean theorem. Furthermore, with our knowledge of the law of cosines we see that the dot product of u and u is equal to (length of u) * (length of u) * Cos(theta). This means that by using the dot product we can find the angle difference of two vectors.

With this knowledge we see that often we have two vectors (ie. u =(2,1) and a=(6,8)) that are not multiples of each other, meaning that we can't multiply the first vector by anything to get the second vector. We want to know in this case, whats the multiple of the first vector that will bring it to be the closest the second. We know that the vectors will be the most similar when the line that passes between the multiplied vector and the desired vector is parallel with the multiplied one. We denote this vector r and it is called the residual vector. According to the dot product u dot r = 0 because they are perpendicular. We want r to be short as possible which means we want it perpendicular. xu + r = a. Because of this we can solve for x to find the multiple that will make the vector closest to the desired one. To do this we set equal:
u dot (xu + r) = a dot r. And knowing that u dot r = 0 because they are perpendicular we solve for x and get in this case x = 4.

2) Challenges

The major challenged I see with this is with the dot product and how to use it algebraically. When solving for x the book distributed u with parenthesis as if it was a multiplication sign, although they are different. I need more clarification on how to use them in equations to avoid mistakes. Other than that it will be important to understand the equations to be able to apply them correctly.

3) Reflections

This seems to be pretty useful considering often there are not solutions to equations, vectors are an easier way to approximate solutions. I hope to see how to apply this vector knowledge to real world stuff.