Mathematics Vectors 1 This set of slides may be updated this weekend. I have given you my current slides to get you started on the PreAssignment. Check back here frequently to see if updates have been made. I will post here a record of updates that have been made.

Mathematics Vectors 2 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

Mathematics Vectors 3 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. Here are some helpful definitions.

Mathematics Vectors 4 N This is an example of a Vector Change Wind Speed Change Wind Direction There are three representations of a vector. 1.Real life: the actual quantity that the vector represents. 2.Mathematical: a number, with units and a direction. 3.Graphical: an arrow which has a length proportional to the magnitude and a direction the same as the vector. Real Life Graphical Representation Mathematical Representation Magnitude Direction Units 24 Northwest mph NortheastSoutheastSouthwest e s w

Mathematics Vectors 5 Change Temperature Graphical Representation Mathematical Representation Magnitude Direction Units 0 none degrees C Degrees C This is an example of a scalar Real Life There are three representations of a scalar as well 1.Real life: the actual quantity that the vector represents. 2.Mathematical: a number, with units and NO direction. 3.Graphical: a point on a graph.

Mathematics Vectors 6 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.

Mathematics Vectors 7 x-axis y-axis AxAx AyAy A θ (the vector) *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! To convert from magnitude/direction to components, we use two equations. (y-component) (x-component) (angle*) (magnitude)

Mathematics Vectors 8 x-axis y-axis 6.43 units 7.66 units 10 units 50 o (the vector) *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! Here is an example. (y-component) (x-component) (angle*) (magnitude)

Mathematics Vectors 9 To convert from components to magnitude/direction, we use two equations. x-axis y-axis AxAx AyAy A θ (the vector) *Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise! (y-component) (x-component) (angle*) (magnitude)

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

Mathematics Vectors 11 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… 1. is a unit vector pointing to the right. 2. is a unit vector pointing up. 3. is a unit vector pointing forward.

Mathematics Vectors 12 Any vector can be written using vector notation. Vector notation uses the special unit vectors. As an example

Mathematics Vectors 13 When adding vectors add their components.

Mathematics Vectors 14 Here is an example of adding vectors.

Mathematics Vectors 15 To add vectors when you are given magnitude/direction, convert to components first.

Mathematics Vectors 16 Here is an example of adding vectors when only their magnitude and direction are given

Mathematics Vectors 17 There are two ways to multiply vectors, but they cannot be divided Dot products produce a scalar. Cross products produce a vector.

Mathematics Vectors 18 When you multiply vectors to get a scalar use a dot product. If you are given the vectors as components (vector notation)…

Mathematics Vectors 19 Here is an example of solving a dot product.

Mathematics Vectors 20 When you multiply vectors to get a scalar use a dot product. *If then subtract it from 360° * If you are given the vectors as magnitude/direction…

Mathematics Vectors 21 Here is another example of solving a dot product. *If then subtract it from 360° *

Mathematics Vectors 22 When you multiply vectors to get a vector use a cross product. If you are given the vectors as components (vector notation)…

Mathematics Vectors 23 When you multiply vectors to get a vector use a cross product.

Mathematics Vectors 24 When you multiply vectors to get a vector use a cross product. *If then subtract it from 360° * Use the right-hand rule to get the direction. If you are given the vectors as magnitude/direction…

Mathematics Vectors 25 When you multiply vectors to get a vector use a cross product. *If then subtract it from 360° Use the right-hand rule to get the direction. *

Mathematics Vectors 26 1.Point the fingers of your right hand in the direction of the vector A. 2.Curl your fingers toward the direction of the vector B. 3.The cross-product vector C is given by the direction of your thumb. Right-Hand Rule

Mathematics Vectors 27 Drawing Vectors 1. Locate the position where the vector is being measured. 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. 3. Label the vector with its name. Put an arrow above the name or make it boldface. 4. If necessary, move the vector to another position, keeping its length and direction the same.

Mathematics Vectors 28 Graphical Vector Addition

Mathematics Vectors 29 x-axis y-axis Negative Vectors

Mathematics Vectors 30 Graphical Vector Subtraction

Mathematics Vectors 31 Graphical Dot Product  cosA

Mathematics Vectors 32 Graphical Cross Product  sin A  A The magnitude of the cross product is the area of a parallelogram that has the two vectors as its sides.

Mathematics Vectors 33 If two vectors form a mirror image around one of the axes, then the component of the resultant along that axis is zero. x-axis y-axis Symmetry