1. Dot Product Second Type of Product Using Vectors

2. Dot Product If v = a 1 i + b 1 j and w = a 2 i + b 2 j are two vectors, the dot product v. w is defined as v. w = a 1 a 2 + b 1 b 2 The answer to a dot product is a number.

3. Properties of the Dot Product If u, v, and w are vectors, then Commutative Property u. v = v. u Distributive Property u. (v + w) = u. v + u. w v. v = ||v|| 2 0. v = 0

4. Angles Between Vectors If u and v are two nonzero vectors, the angle θ, 0 ≤ θ ≤ , between u and v is determine by the formula

5. Finding the Angle between Two Vectors Example

6. Navigation Problems Finding the Actual Speed and Direction of an Aircraft Example page 632 On-line Examples

7. Parallel and Orthogonal Vectors Two vectors are said to be parallel if the angle between the two vectors is 0 or  Two vectors are orthogonal (at right angles), if the angle between the two nonzero vectors is  / 2 or the dot product is 0.

8. Projection of a Vector onto Another Vector or Decomposition Vector Projection allows us to find “how much” of the magnitude is working in the horizontal direction and “how much” is working in the vertical direction. We decompose the one vector into a vector that is parallel to the vector we are projecting onto and one that is orthogonal to the vector we are projecting onto.

9. Vector Projection Remember that we will always have two vectors when we are through. If v and w are two nonzero vectors, the vector projection of v onto w is

10. Decomposition of v into v 1 and v 2 The decomposition of v into v 1 and v 2, where v 1 is parallel to w and v 2 is perpendicular to w, is

11. Work Done by a Constant Force Work = (magnitude of force) (distance) Up till now all work you have been computing has been at an angle of 90 degrees or 0 degrees. Vectors allow us to push or pull at any angle.

12. Work Done by a Constant Force Work done by a force using vectors is computed as