AP Physics C: Intro to Vectors



AP Physics C: Intro to Vectors

Vector- a quantity that has magnitude and direction. The magnitude of a vector can never be less than zero and mist be given with units.

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Time for some new notations. Consider a vector that has a magnitude of one- This is a unit vector! You can make something a unit vector by normalizing it- take the vector and divide by its magnitude.

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Write this vector in terms of unit vector notation:

What is the magnitude and direction of this vector?

What we did last year- Vectors can be added (subtraction is just adding a vector rotated 180 degrees) and we learned that you can do this graphically or analytically.

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Analytically: You can break each vector down into components- most often along the coordinate axes

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What we will do this year- Multiplication!!! The Dot and Cross Product of two vectors as well as the scalar multiplied by a Vector

THE EASY ONE- SCALAR MULTIPLIED BY A VECTOR- WILL RESULT IN VECTOR QUANTITY

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MULTIPLICATION: THE DOT PRODUCT – How parallel are two given vectors?

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The dot product is commutative, associative, and distributive + THE DOT PRODUCT OF TWO VECTORS ALWAYS YIELDS A SCALAR!!!

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Applications Using New Vector Notations:

1.1 At 2 am one morning you decide to take a jogging run through the MIT buildings. You take 54 seconds to run 250 m along the infinite corridor at MIT from Mass Ave to the end of Building 8, you turn right at the end of the corridor and take 42 seconds to run 178 m to the end of Building 2, and then you turn right, run down the hall for 9 seconds covering 30 m until you to stop.

a) Construct a vector diagram that represents your motion. Indicate your choice of unit vectors.

b) What are the directions and magnitudes of your average velocity for each leg of your trip?

c) What is the direction and magnitude your average velocity for the entire trip?

3 .1 A Coast Guard ship is located 35 km away from a checkpoint in a direction 42 degrees north of west.

A distressed sailboat located in still water 20 km from the same checkpoint in a direction 36 degrees south of east is about to sink. Draw a diagram indicating the position of both ships. In what direction and how far must the Coast Guard ship travel to reach the sailboat?

Cross Products- How perpendicular are two vectors?

The magnitude of the cross product represents the area of the parallelogram defined by the two vectors (each triangle in the parallelogram would have the areas of ½ AB sinθ) in the plane of both vectors. This type of multiplication always results in a vector perpendicular to the plane that the other two vectors lie in.

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The cross product is not commutative!!!

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To find the cross product of two vectors, you can use a method that uses determinants for finding the actual x,y,z components:

An application is to find the torque on a rigid body. For example:

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Another application is Magnetic force on a charged particle in a B-Field.

Time to practice- Problem 5 from end of Chapter USED MATH and then

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