1. 25. Dot Product of Vectors

2. Dot Product

3. Example

4. Example

5. Example

6. Applications of dot product
Finding the angle between vectors
Determining if vectors are orthogonal (perpendicular) or parallel
Projecting a vector onto another
Work

7. Angle Between Two Vectors
This formula comes from the law of cosines!!

8. Example

9. Example

10. Orthogonal Vectors Parallel Vectors
The vectors u and v are orthogonal if u·v = 0.
Parallel Vectors
The vectors u and v are parallel if u·v =

11. Example
Are vectors u = <2,-3> and v = <6,4> orthogonal, parallel, or neither?
Orthogonal

12. Example – Tell if vectors are orthogonal, parallel, or neither

13. Projecting a vector onto another
We have seen applications of finding a resultant vector such as forces pulling on an object or wind resistance on a plane
There are other applications in physics and engineering where you need to do the reverse – decompose the vector into the sum of 2 perpendicular vector components

14. Vector components
Consider a boat on an inclined ramp shown below. The force F due to gravity pulls the boat down the ramp (w1) and against the ramp (w2) .
Notice that w1 and w2 are orthogonal. These are called vector components.

15. To find w1 and w2 (the vector components)
w1 is the projection of u onto v and is denoted w1=projvu
w2 = u - w1
The projection (w1) is like
shining a light onto a vector
and finding its shadow on the
second vector

16. Example
Find the projection u = <3,-5> onto v = <6,2>. Then write u as the sum of two vector components.

17. Work
Work is the product of the force acting in the direction something moves and the distance it travels
There are 2 ways to find this – we will only look at one – the dot product
Find the component form of the force and the component form of the displacement and then find their dot product
W =

18. Example
Find the work done by a force F = 4i + 9j in moving an object from (4,6) to (8, 7)

19. Example
Find the work done by a force of 53 N at 47o in moving an object 36 m horizontally.