1. Lesson 6.4 – Dot Products The dot product of two vectors is given by
The result is a scalar (number) – not a vector

2. The dot product has all the properties of multiplication

3. 13 Answer Example: Find the dot product FYI…
Mechanical work is the dot product of force and displacement vectors.
Magnetic flux is the dot product of the magnetic field and the area vectors

4. This is a dot product since u and v are vectors
Angle Between 2 Vectors
You could find the angle between the vectors using a 2 step process:
Find the direction angle of u v u
Find the direction angle of v
Find the difference
Or with 1 step:
This is a dot product since u and v are vectors

5. Example Find the angle between the vectors
Answer
104.6

6. Orthogonal Vectors
All vectors can be written as the sum of 2 perpendicular vectors: u w2 w1
Perpendicular vectors are also called orthogonal, and their dot product = 0

7. Orthogonal Vectors can be used to find the projection of a vector onto another
Projection: The force and direction of a vector that balances (equals) another vector
The projection of u onto v is:
What vector, v, is needed to keep the wagon from rolling back if F is 300 pounds?
F = -300j F v
30o
v = cos 30oi + sin 30oj