- Topics At a Glance
- Functions
- Increasing or Decreasing or...
- Bounded
- Even and Odd Functions
**Vectors: A New Kind of Animal****Magnitude**- Direction
- Scaling Vectors
- Unit Vectors
- Vector Notation
- (More than 2)-Dimensional Vectors
- Vector Functions
- Sketching Vector Functions
- Parametric Equations
- Graphing Parametric Equations
- Points on Graphs of Parametric Equations
- Parametrizations of the Unit Circle
- Parameterization of Lines
- Polar Coordinates
- Simple Polar Inequalities
- Switching Coordinates
- Translating Equations and Inequalities between Coordinate Systems
- Polar Functions
- Graphing Polar Functions
- Rules of Graphing We Do (or Don't) Have
- Bounds on Theta
- Intersections of Polar Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

With an understanding that vectors are simply mathematical equivalents of mythological hybrid beasts, we will investigate their parts to see if we can understand them better. Surely, with the help of the minotaur and griffin vectors, the calculus bear stands no chance against you, right?

The magnitude of a vector is like the body of the vector. We know *how much* vector is there. We use the notation

||

When we draw a vector *xy*-plane, the length of the arrow is the magnitude of the vector.

*x* and *y*. We can find the magnitude of the vector using the Pythagorean Theorem.

**Be Careful:** When finding the magnitude of a vector with negative components, either

use parentheses and square the components correctly, or

drop the negative signs altogether.

Since finding magnitude requires squaring the components anyway, it doesn't matter if the components are negative or positive.

Magnitude is a lot like absolute value in this way.

Whether a number is positive or negative doesn't affect its absolute value: |-5| = |5|.

Similarly, whether the components of a vector are positive or negative doesn't affect its magnitude: ||<1,2>|| = ||<-1,2>|| = ||<1,-2>|| = ||<-1,-2>||.

Look at the vectors on a graph. The arrows are all the same length; they're pointing different directions:

If a vector lies flat along the *x- *or *y-*axis, we don't need to use the Pythagorean Theorem to find its magnitude. Such a vector will have only one component that isn't zero, and that component will be the magnitude of the vector.

Example 1

What is the magnitude of the vector <3,4>? |

Example 2

Find the magnitude of the vector <-2,3>. |

Example 3

Find the magnitude of the vector <0,7>. |

Exercise 1

Find the magnitude of each vector.

- <1,1>

Exercise 2

Find the magnitude of each vector.

- <-2,4>

Exercise 3

Find the magnitude of each vector.

- <-5,-6>

Exercise 4

Find the magnitude of each vector.

- <4,-7>

Exercise 5

Find the magnitude of each vector.

- <5,0>