Definitions

1. Definitions Examples of a Vector and a Scalar More Definitions Components, Magnitude and Direction Unit Vectors and Vector Notation Vector Math (Addition, Subtraction, Multiplication) Drawing a Vector Graphical Vector Math Symmetry Sample Problems

2. Here are some helpful definitions. Magnitude: The amount of a quantity represented by a vector or scalar. Direction: The angle of a vector measured from the positive x-axis going counterclockwise. Scalar: A physical quantity that has no dependence on direction. Vector: A physical quantity that depends on direction. Units: A standard quantity used to determine the magnitude of a vector or value of a scalar.

3. This is an example of a Vector Change Wind Speed • There are three representations of • a vector. • Real life: the actual quantity that the vector represents. • Mathematical: a number, with units and a direction. • Graphical: an arrow which has a length proportional to the magnitude and a direction the same as the vector. Change Wind Direction Graphical Representation N Mathematical Representation Real Life 6 24 18 12 Magnitude w e Northwest Northeast Southeast Southwest Direction Units mph s

4. This is an example of a scalar. • There are three representations • of a scalar as well • Real life: the actual quantity that the vector represents. • Mathematical: a number, with units and NO direction. • Graphical: a point on a graph. Change Temperature Real Life Graphical Representation Degrees C Mathematical Representation Magnitude 100 75 0 25 50 Direction none Units degrees C

5. More Definitions Component: The projection of a vector along a particular coordinate axis. Dot Product: The product of two vectors the result of which is a scalar. Cross Product: The product of two vectors the result of which is another vector. Right-Hand Rule: The rule which gives the direction of a cross-product.

6. To convert from magnitude/direction to components, we use two equations. y-axis *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! (magnitude) Ay A (y-component) (the vector) θ (angle*) x-axis Ax (x-component)

7. Here is an example. y-axis *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! (magnitude) 10 units 7.66 units (y-component) (the vector) 50o (angle*) x-axis 6.43 units (x-component)

8. To convert from components to magnitude/direction, we use two equations. y-axis *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! (magnitude) Ay A (y-component) (the vector) θ (angle*) x-axis Ax (x-component)

9. Here is an example. y-axis *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! (magnitude) 10 units 7.66 units (y-component) (the vector) 50o (angle*) x-axis 6.43 units (x-component)

10. A unit vector is any vector with a magnitude equal to one. To find a unit vector in same direction as the vector, divide the vector by its magnitude. • There are three special unit vectors… • is a unit vector pointing to the right. • is a unit vector pointing up. • is a unit vector pointing forward.

11. Any vector can be written using vector notation. Vector notation uses the special unit vectors. As an example

12. When adding vectors add their components.

13. Here is an example of adding vectors.

14. To add vectors when you are given magnitude/direction, convert to components first.

15. Here is an example of adding vectors when only their magnitude and direction are given

16. There are two ways to multiply vectors, but they cannot be divided Dot products produce a scalar. Cross products produce a vector.

17. When you multiply vectors to get a scalar use a dot product. If you are given the vectors as components (vector notation)…

18. Here is an example of solving a dot product.

19. When you multiply vectors to get a scalar use a dot product. If you are given the vectors as magnitude/direction… * *If then subtract it from 360°

20. Here is another example of solving a dot product. * *If then subtract it from 360°

21. When you multiply vectors to get a vector use a cross product. If you are given the vectors as components (vector notation)…

22. When you multiply vectors to get a vector use a cross product.

23. When you multiply vectors to get a vector use a cross product. If you are given the vectors as magnitude/direction… * Use the right-hand rule to get the direction. *If then subtract it from 360°

24. When you multiply vectors to get a vector use a cross product. * Use the right-hand rule to get the direction. *If then subtract it from 360°

25. Right-Hand Rule • Point the fingers of your right hand in the direction of the vector A. • Curl your fingers toward the direction of the vector B. • The cross-product vector C is given by the direction of your thumb.

26. Drawing Vectors 1. Locate the position where the vector is being measured. 3. Label the vector with its name. Put an arrow above the name or make it boldface. 2. Draw an arrow, with a tail at the vector position, pointing in the direction of the vector and having a length proportional to its magnitude. 4. If necessary, move the vector to another position, keeping its length and direction the same.

27. Graphical Vector Addition

28. Negative Vectors y-axis x-axis

29. Graphical Vector Subtraction

30. f A cos Graphical Dot Product

31. f f A A sin sin Graphical Cross Product The magnitude of the cross product is the area of a parallelogram that has the two vectors as its sides.

32. Symmetry If two vectors form a mirror image around one of the axes, then the component of the resultant along that axis is zero. y-axis x-axis

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