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What are vectors and why do we need them?

Scalars are single numbers with units attached to them. Vectors are a scalar coupled with a direction. Some examples of vector quantities include velocity (speed + direction) and displacement (distance traveled + direction). The scalar quantity in a vector is referred to as the magnitude. (Btw from past experience I know that vectors do not necessarily have to have a direction, they're just multiple scalars coupled together. I'm sure they're just explaining it like this right now because that's what we're gonna use mostly for this part of the book).

Terminology and notation

  • Vectors can be represented symbolically as a bold-faced letter with a right-facing horizontal arrow on top (to signify direction) and (usually) in all-caps. In handwritten form this is the same thing but instead of being bold-faced it just has a line under the letter.
  • Parallel vectors are vectors that have the same direction.
  • You can represent the magnitude of a vector by just typing it without bold-face or surrounding it with two vertical lines.
  • The negative of a vector is a vector with the same magnitude but the exact opposite direction.
  • Vectors that have the exact opposite direction (whether or not the magnitudes are the same) are referred to as being anti-parallel.
  • The resultant is the sum of two vectors

Vector Arithmetic

Vector addition works by connecting vectors in the order they're being added by from head to tail or by connecting their tails and forming a parallelogram. Tbh I prefer the first because you can do it more easily if you have more than two vectors being added whereas with the second one you have to do two at a time then take the resultant and add it to the next vector, etc. A much more laborious process. Vector addition does not usually mean just adding the magnitudes. Usually this gives a very incorrect answer. Also, order of terms being added doesn't matter (aka they're commutative). Vector subtraction works by adding the first vector to the negative of the second vector. Multiplying a vector by a scalar just multiplies the magnitude of that vector by the absolute value of that scalar. And if that scalar is negative, you just flip the direction of the vector.

Vector components

The more general form of performing vector arithmetic is by breaking down a vector into its components. A component is simply the different coordinates that make up the position of the head of a vector. If you are given the magnitude and the direction (aka angle of a vector relative to some reference direction), you can find the components by using the definitions of the sine and cosine functions (think SOH CAH TOA). If you're given the components, you can find the magnitude via the Pythagorean theorem. Adding two vectors just means adding their corresponding components to one another. You can also find the direction of a vector if you have the components by using definition of the tangent function.

Unit vectors

A unit vector is any vector that has a magnitude of 1 but with no unit attached to it. Its only purpose is to point in space. They are usually represented by a lower case letter with a little carret ("hat") above it.

Vector Products

There are two main types of products of vectors: the scalar product (produces a scalar) and the vector product (produces a new vector).

Scalar Product

(Also called the dot product because of the notation used to represent the scalar product).

  1. Place the two vectors tail-to-tail
  2. Find the angle between them and call it ϕ\phi
  3. Multiply the magnitudes of each vector together and multiply that by the cosine of ϕ\phi: ABcos(ϕ)AB\cos(\phi). (Might be wise to make an interactive animation here that demonstrates the geometric meaning of the scalar product. Not sure I fully grasp it)

Vector Product

Also called the cross product because of the notation used to denote the operation.

  1. Place the two vectors tail-to-tail.
  2. Find the angle between them and call it ϕ\phi.
  3. Multiply the magnitudes of each vector together and multiply that by the sine of ϕ\phi: ABsin(ϕ)AB\sin(\phi). This gives the magnitude of the new vector generated. I really am not sure I understand this stuff because I don't see any direct correspondance to physical reality yet and why these operations are defined this way.