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.

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 ________________

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

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).

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.

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

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).

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.

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.

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

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:

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

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.

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

Direction Angles

Finding the components of a Vector

Finding the Direction Angle of a Vector

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