1. Scalars
Some quantities, like temperature, distance, height, area, and volume, can be represented by a ________________ that indicates __________________, or _____________.
These quantities are called a ________________.
For example, the quantity “_______________________” is a regular number, or scalar.

2. What are Vectors?
The quantity “60 miles per hour _________________________” is a called a ___________, because it has both _________ and ___________________.
Vectors are a quantity that have both magnitude (length) and direction, usually represented with an __________:
This includes _______, __________, and ________________

3. Vectors
Two vectors are equal if they have the same ______________ and ___________.
We denote vectors by lowercase boldface letters such as _____________, and so forth.
A vector can also be written as the letters of its head and tail with an arrow above v

4. Component Form of a Vector
If v is a vector in the plane equal to the vector with initial point (0, 0) and terminal point (v1, v2), then the component form of v is
v = _________
The numbers v1, and v2 are the __________ of v.
The vector <v1, v2> is called the ____________ vector of the point (v1, v2).

5. Finding the Component Form
If the initial point of a vector, u, is not (0, 0), we can find the components of u by ___________ the x- and y-coordinates of the initial point from the terminal point.
Example: If u is the vector with initial point P(1, 2) and terminal point Q(5, 7), find the component form of u.

6. Finding Magnitude of a Vector
The magnitude (or length) of a vector v is denoted by vertical bars on either side of the vector: _____
OR it can be written with double vertical bars: ____
Finding Magnitude of a Vector
If v is represented by the arrow from

7. Finding the Component Form and Magnitude of a Vector
Find the component form and magnitude of the vector 𝒗= 𝑃𝑄 , where P = (-3, 4) and Q = (-5, 2).

8. Showing Vectors are Equal
Let u be the vector represented by the directed line segment from R to S, and v the vector represented by the directed line segment from O to P. Prove that u = v.

9. Vector Operations To add vectors, simply ___________________
Two basic vector operations are vector addition and scalar multiplication.
To add vectors, simply ___________________
To multiply a vector by a scalar, simply ______ _________________________by the scalar.

10. Examples
If u = <-1, 3> and v = <4, 7>, find the component form of:
u + v =
3v =
2u + (-1)v =

11. Unit Vectors
A _______________ is a vector u with magnitude (length) of 1.
Given a vector v, we can form a unit vector by multiplying the vector by:

12. Remember: The magnitude of a Unit Vector should be 1
Finding a Unit Vector
Remember: The magnitude of a Unit Vector should be 1

13. Standard Unit Vectors
The two unit vectors _________ and ________ are the ______________ unit vectors.
Any vector v can be written as an _____________ in terms of the standard unit vectors.
Example: Write the vector <3, 4> as a linear combination of the standard vectors.

14. Direction Angles
The precise way to specify the direction of a vector is to state its ____________________________, the angle 𝜃 that v makes with the positive x-axis. v

15. Direction Angles

16. Finding the components of a Vector

17. Finding the Direction Angle of a Vector

18. Finding the direction angle
Find the direction angle for the vector <8, -4> v