Vectors and Dot Product 6.4 JMerrill, 2010

Quick Review of Vectors: Definitions Vectors are quantities that are described by direction and magnitude (size). Example: A force is a vector because in order to describe a force, you must specify the direction in which it acts and its strength. Example: The velocity of an airplane is a vector because velocity must be described by direction and speed. The vector is the zero vector. It has no direction.

Representation of Vectors The velocities of 3 airplanes, two of which are heading northeast at 700 knots, are represented by u and v u v W e say u = v to indicate both planes have the same velocity. w ≠ u or v because the direction is different. w

Magnitude The magnitude of vector v is represented by the absolute value of v (with double bars) In the previous example, We know that, but u ≠ w, and v ≠ w because the direction is not the same.

Addition of Vectors A B C 10 5 The sum is called the resultant.

Component Form From the tail to the tip of vector v, we see: A 2 unit change in the x- direction, and A -3 unit change in the y- direction. 2 and -3 are the components of v. When we write v = we are expressing v in component form. 2 3

You can count the number of spaces to get the component form or, you can subtract the coordinates. IT IS ALWAYS B – A! The magnitude of vector AB is found using the distance formula: (x 1,y 1 ) (x 2,y 2 ) (x 2 – x 1 ) (y 2 – y 1 )

Example Given A(4, 2) and B(9, -1), express in component form. Find

Vector Operations with Coordinates VVector Addition vv + u = VVector Subtraction vv - u = SScalar Multiplication kk v =

Example If u = and v =, find: u + v u – v 2 u – 3 v

Definition – Dot Product Example: Find the dot product: 6 (-2) + (1)(3) = = -9 The dot product gives you a scalar, NOT a vector!

Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar.

Angle Between Two Vectors

Example Find the angle between

Orthogonal (Perpendicular) Vectors Two vectors are orthogonal if their dot product is 0 Example:

You Try Find Find Find the angle between