2007-09-20 - Vectors
Today's nonsense
- Dakota's watchlessness and anxiety
- Bryan's new position as the "math person" of the math club
- Dakota and Bryan running at each other, example
This is chapter 3. Vectors represent quantities that have magnitude and direction. So mangitude can be stuff like size, it has a value, or speed. The direction makes the value either positive or negative. Component form of vectors, well, we can talk about how we can arrange it so that the vector starts at the origin, and the magnitude is just the length, and the direction would be the angle. We're going to try to answer all of that. So what are some quantities that are vectors?
Velocity, acceleration, displacement, force, momentum, --- quantities with magnitude but with no direction are going to be scalars.
Scalars: speed, distance.
Lockhart overdoes the visual addition of vectors.
Graphically
Graphically they are represented by arrows (rays, but rays go on forever, and that's not right for vectors), or "line segments with arrowheads on one end." The arrows indicate direction and whose length is proportional to its magnitude.
Two vectors can be added graphically by placing the tail of one to the head of the other. The resultant, will be the connection from the beginning of the first one to the head of the second one that has been visually added on (this could be done algebraically with the Pythagorean theorem or distance formula, of course). There's also the parallelogram method where the diagnol of the parallelogram would be the, yeah.
So let me ask you. Now, vector c = vector a + vector b, so you need to realize, or understand what this statement says, this does not apply that C = A + B, because the magnitudes are different, so when you do not mention that it is a vector, it defaults to magnitude.
The Triangle Rule: The sum of any two legs is always more than the third leg.
The Parallelogram Method
The parallelogram method is another way to look at adding vectors. Place the two vectors tail to tail. The diagonal of the parallelogram formed by VectorA and VectorB. equals VectorC. So take VectorA, and remember VectorB, and this says that you put them tail to tail, and so here's VectorA and here's VectorB, and then make a parallelogram out of them, so that VectorC is the vector splitting B & A so that it meets from the intersection of vectors A&B and then goes to the imaginary intersection of the two other sides.
Two vectors are equal if they have the same magnitude and same direction. This means they have the same length and are parallel. Moving a vector so that it remains parallel to itself does not change it. So that's why we can do this graphical technique of adding vectors.
Tho subtract vectors graphically we add the opposite vector that you have to subtract. So VectorA - VectorB = VectorA + ( - VectorB). Opposite vectors have the same magnitude but opposite direction.
An equivalent way of subtracting VectorB from VectorA is to draw them tail-to-tail and then draw a VectorC from VectorB to VectorA. In this scenario, VectorC must be added to VectorB to find VectorA). So VectorA - VectorB = VectorC so VectorA=VectorB + VectorC.
Vectors can easily be added algebraically, rather than graphically, by using the components of vectors, their projected horizontal and vertical lengths.