1. Math 200
Week 1- Wednesday
Vectors

2. Main Questions for Today
Math 200
Main Questions for Today
What is a vector?
What operations can we do with vectors?
What kinds of representations and notations do we have for vectors?
What are some of the important properties of vectors?

3. Math 200
Definition
A vector is a mathematical object with magnitude and direction.
Representation: arrow whose length corresponds to magnitude
Notation:
Bold letters - e.g. v as opposed to v
Arrow over letter:

4. Math 200
Sums of vectors
We add two vectors by putting them tail-to-end and then drawing the resultant vector from start to finish.
E.g., say we want to add two vectors v and w, shown below.

5. Math 200
¡¡Important!!
VECTORS DO NOT HAVE FIXED POSITION!
THESE ARE ALL THE SAME BECAUSE THEY HAVE THE SAME DIRECTION AND MAGNITUDE

6. Scalar Multiples of vectors
Math 200
Scalar Multiples of vectors
A scalar is a quantity with only magnitude (no direction)
By multiplying a vector by a scalar, we can
Scale its length/magnitude
Change its direction if the scalar is negative

7. Differences of vectors
Math 200
Differences of vectors
With scalars, we can think of subtraction as addition of negative numbers.
E.g., = 5 + (-2) = 3
The same goes for vectors: v - w = v + (-w)

8. Vectors in rectangular Coordinates
Math 200
Vectors in rectangular Coordinates
Suppose I draw a vector from one point to another. Call these points P1(x1, y1, z1) and P2(x2, y2, z2) P2
We’ll write this vector in the following way: P1

9. Math 200
Standard Position
When we draw a vector with its tail at the origin, we’ll say it’s in standard position.
For a vector in standard position, the terminal point gives the components directly…
So if this is the point (2,5,2), then the vector is…

10. A 2d Example Do the following: Draw and label x and y axes
Math 200
A 2d Example
Do the following:
Draw and label x and y axes
Pick two vectors (e.g. v = <1,2> and w = <-3,1>)
Draw v with its tail at the origin
Draw w with its tail at the terminal point of v
Draw v + w

11. Math 200
<1+3, 2+1> = <4,3>
<3,1>
<1,2>

12. properties Vector operations work pretty much the way we’d like!
Math 200
properties
Vector operations work pretty much the way we’d like!

13. Math 200
Parallel Vectors
We saw that scalar multiplication always gives us a parallel vector to the original.
E.g. If v = <1,2,-4>.
Then 2v = <2, 4, -8> and -3v = <-3, -6, 12> are both parallel to v
Conversely, we can say that two vectors are parallel if they are scalar multiples of one another!
E.g. v = <1,3,1> and w = <-4, -12, -4> are parallel because w = -4v
Non-example: v = <1,3,1> and q = <2, 6, 4> are not parallel.
How do we know? q1 = 2v1 but q3 = 4v3

14. Norm of a vector The norm of a vector is just its magnitude/length.
Math 200
Norm of a vector
The norm of a vector is just its magnitude/length.
We write ||v|| for the “norm of v”
Since vectors don’t have a fixed position, we can put any vector in standard position and use those components in the distance formula:
E.g.,

15. What can we say about ||kv||? We know that k scales the magnitude of v
Math 200
What can we say about ||kv||?
We know that k scales the magnitude of v
We also know that if k is negative, we flip the direction of v
Let’s see if that intuition corresponds to the algebra:

16. Unit vectors When a vector has length one, we call it a unit vector
Math 200
Unit vectors
When a vector has length one, we call it a unit vector
Later, we’ll use unit vectors to define something called the directional derivative
Often, we’ll need to take a non-unit vector, and shrink it to unit length
This is called normalizing a vector

17. Math 200
Normalizing vectors
Suppose we want a vector in the same direction as a vector v = <2,-1,1>, but with length one.
We know that scalar multiplication by a positive number preserves direction.
Let’s multiply v by 1/||v||

18. Three special unit vectors
Math 200
Three special unit vectors
The three simplest unit vectors: <1,0,0>, <0,1,0>, <0,0,1>
These get special names:
i = <1,0,0>
j = <0,1,0>
k = <0,0,1>
An alternative notation for vectors uses i, j, k:
E.g., <2,3,6> = 2i + 3j + 6k
And sometimes they get hats

19. Math 200
An Application
<Acos(3π/4),Asin(3π/4)>
<Bcos(π/6),Bsin(π/6)>
*Think of each of these vectors the hypotenuse of a right triangle. We want to write the x and y components.
<0,-200>

20. If two vectors are equal, they must have the same components.
Math 200
If the weight isn’t moving, the three force vectors should add up to <0,0>
If two vectors are equal, they must have the same components.

21. Math 200
Elimination
Substitution