1. 24. Dot Product of Vectors

2. What you’ll learn about  How to find the Dot Product  How to find the Angle Between Vectors  Projecting One Vector onto Another … and why Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

3. Dot Product

4. Example Finding the Dot Product

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

8. Example Finding the Angle Between Vectors

10. Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0. The terms orthogonal and perpendicular mean essentially the same thing – meeting at a right angle.

11. Example  Are vectors u = and v = orthogonal?  Find the dot product.  Therefore, yes. If you graph, you will see a right angle.

12. Parallel vectors  Two vectors are parallel if  Dot product = negative product of magnitude

13. Example – Tell if vectors are perpendicular, parallel, or neither

14. Finding vector components  There are many applications in which 2 vectors are added together to find a resultant vector such as forces pulling on an object or wind resistance on a plane  There are many applications in physics and engineering where you need to do the reverse – decompose the vector into the sum of 2 vector components

15. Definition of vector components  Let u and v be nonzero vectors such that u = w 1 and w 2 Where w 1 and w 2 are orthogonal and w 1 is parallel to v.  w 1 and w 2 are called vector components

16. To find w 1 and w 2 (the vector components)  W 1 is the projection of u onto v and is denoted W 1 =proj v u  W 2 = u - w 1

17. Projection of u onto v

18. Example  Find the projection u = onto v =. Then write u as the sum of two vector components.