1. Section 6.7 The Dot Product

2. Overview From last section, adding two vectors results in another vector. So also does multiplying a scalar (real number) by a vector. In this section we explore an operation between vectors that results in a scalar.

3. Givenand Define the dot product as

4. Examples Find the dot product for the following pairs of vectors: 1.v = 9i + j and w = i + 9j 2.v = 3i – 4j and w = 7i – 2j Let Find

5. The second formula… …can be used to find the angle between two vectors.

6. Examples Find the angle between the two given vectors. 1.v = 2i and w = 2i + 3j 2.v = 9i + 4j and w = 5i – j

7. Orthogonal Vectors Two vectors are orthogonal (meet at a 90- degree angle) if and only if the dot product between the two vectors is equal to 0. Two vectors are parallel if the angle between the vectors is either 0° or 180°. Strategy: find the dot product. If it is 0, the vectors are orthogonal. If not, go on to find the angle between the vectors, keeping in mind that cos 0° = 1 and cos 180° = -1.

8. Parallel, Orthogonal, or Neither? 1.v = 7i +63j and w = 9i – j 2.v = 2i – 7j and w = -10i + 2j 3.v = 4i + 8j and w = 4i – 2j 4.v = 3i – 2j and w = 6i + 12j

9. Work The work W, done by a force F moving an object from A to B is given by Where F is the force moving an object from A to B, AB is the distance over which the constant force is applied, and “theta” is the angle between the force and the direction of motion.

10. Examples A wagon is pulled along level ground by exerting a force of 26 pounds on a handle that makes an angle of 45° with the horizontal. How much work is done pulling the wagon 90 feet? Find the work done in pushing a car along a level road from point A to point B, 91 feet from A, while exerting a constant force of 91 pounds.