There are two types of multiplication that can be done with vectors: = +.

Transcription

1 Section 7.5: The Dot Product Multiplying Two Vectors using the Dot Product There are two types of multiplication that can be done with vectors: Scalar Multiplication Dot Product The Dot Product of two vectors =, and =, is given by =+. Note that and need to be the same size of vector and the result of the dot product is a scalar. Example 1: Finding the Dot Product of Two Vectors Find the dot product of 1,2 3,4. Properties of the Dot Product 1. = 2. = 3. =0 4. = = 5. + = = +

2 How could we find the angle between two vectors? If is the angle, 0 180, between two nonzero vectors and, then = Example 2: Finding the Angle Between Two Vectors Find the smaller angle between 2, 1 and 1,3. When two vectors are parallel, the angle between them is When two vectors are perpendicular, the angle between them is Suppose two non-zero vectors are perpendicular, meaning that 90 = Then:

3 Orthogonal Vectors Two vectors and are orthogonal (perpendicular) if and only if their dot product is zero. Example 3: Determining Whether Vectors are Orthogonal Determine whether each pair of vectors is an orthogonal pair. a. = 6,8 and = 8, 6 b. = 6, 4 and = 6, 9 Work! Work is done when a force causes an object to move a certain distance. Simplest case: When force is going in the same direction as the displacement - Think a stagecoach with horses, the horses pull with a force in the same direction. In this case, =, or the magnitude of force over a distance,, and this would result in being a scalar.

4 However, life is cruel, and often times force will not be in same direction as displacement, so more often than not, we ll need more than just the magnitude of the vector. Suppose my truck ran out of gas and I m stuck having to push it to the nearest gas station. Part of the force, that I m using to push my truck is going towards forward, horizontal motion, or. Suppose I let be the angle between these two force vectors. Then I could solve for =. Wee little example: If I push on my truck with 150 pounds of force with an angle of 30, the horizontal component of the force vector is This means that our equation for work from earlier would be in general = = Where d is the displacement vector and F is the force vector.

5 Example 4: Calculating Work How much work is done when a force (in pounds) = 2,4 moves an object from 0,0 to 5,9 (where distance is in feet). Example 5 (#50) To slide a crate across the floor, a force of 800 lbs at a 20 angle is needed. How much work is done if the crate is dragged 50 feet? Round to the nearest ft-lb.

6 Chapter 8 Section 8.4: Polar Equations and Graphs Polar Coordinates The Polar Coordinate System is centered around the pole, which corresponds to the origin on the Cartesian Plane. Additionally, there is the polar axis which is a ray with its endpoint at the pole. Angles are then labeled around the entire graph either in radians or degrees. How to plot polar coordinates: To plot the point,: 1. Start on the polar axis and rotate the terminal side of the angle to the value. 2. If >0, the point is units from the origin in the same direction of the terminal side of. 3. If <0, the point is units from the origin in the opposite direction of the terminal side of.

7 Example 1: Plotting Points in the Polar Coordinate System Plot the following points in the polar coordinate system. a. 3, ) b. 2,60 Converting Between Polar and Rectangular Coordinates From To Identities Polar, Rectangular, = and = Rectangular, Polar, = + and =,0

8 Example 2: Converting Between Polar and Rectangular Coordinates a. Convert the rectangular coordinate 1, 3 to polar coordinates. b. Convert the polar coordinate 3 3,75 to rectangular coordinates. Graphs of Polar Equations Not only do we have graphs of equations on the coordinate plane, but there are also (pretty!) graphs of polar equations. Example 3: Graphing a Polar Equation of the Form = or = Graph the polar equations a. =3 b. =

9 Example 4: Graphing a Polar Equation of the Form = or = Graph =4. =4,

10 Example 5: Graphing a Polar Equation of the Form cos2 or sin 2 Graph 5sin2 5sin 2, Let s use our calculators to graph the following: Example 6: The Cardioid as a Polar Equation Graph 22 Example 7: Graphing a Polar Equation of the Form Graph /2

11 In some cases, plotting an equation in the form that it is given in is not convenient. For example, plotting than the same equation in rectangular form. Example 8: Graph. is much more complicated