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Lesson 2 Vectors

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    1. Lesson 2 Vectors

    2. Vectors - Definition A vector is a linear independent combination of components with associated basis vectors. For example The components are The associated basis vectors are

    3. Components and Basis vectors The components will change depending on the Coordinate System (CS) The basis vectors indicate the choice of CS. For most cases in this course we will assume an orthogonal rectangular CS where - east-west direction - north-south direction - vertical or z direction

    4. Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions on how to label a point relative to the origin and the axes

    5. r is the hypotenuse and ? an angle ? must be ccw from positive x axis for these equations to be valid

    6. Scalar Magnitude Vector Magnitude Direction Vector representation An arrow that points in the quantitys direction Length is proportional to the magnitude

    7. Notation There are 4 common ways to represent vectors 1. Arrow notation: 2. Boldface: A 3. Underline: 4. Index notation

    8. Vector Decomposition

    10. Components of a Unit Vector Unit vector Dimensionless Magnitude = 1 Direction = along the coordinate axis The symbols: Ax is the same as Ax and Ay is the same as Ay etc. The complete vector can be expressed as

    11. Vector properties In order to determine if a solution or mathematical object is a vector or scalar, one must recall the original definition of a vector as well as understand the various properties of the vector. 1. Two vectors added together produce a new vector For vectors and The addition of these two vectors leads to a new vector Vector addition is both associative and commutative What are the components of the vector ?

    12. When adding vectors, their directions must be taken into account Units must be the same All of the vectors must be of the same type of quantity For example, you cannot add a displacement to a velocity Graphical Methods Use scale drawings Algebraic Methods More convenient

    13. Vector Addition

    16. Adding Vectors Using Unit Vectors Using R = A + B Then and so Rx = Ax + Bx and Ry = Ay + By

    17. Subtraction of Vectors

    18. Vector properties 2. A scalar quantity can be multiplied with a vector to produce a vector For vector and scalar quantity Scalar multiplication leads to a new vector What are the components of ?

    19. Multiplication by a Scalar From a geometric standpoint, scalar multiplication causes a stretching or compression to the length of the vector. If then vector is stretched If then the vector is compressed If then vector is in opposite direction

    20. Vector properties 3. The scalar or dot product is a way to multiply two vectors together such that the result is a scalar. For vectors The dot product is defined algebraically as

    21. Vector properties - Dot Product There also is a geometric definition: Where is the co-planar angle between and By the dependence on we can see the dot product is a measure of how parallel two vectors are to each other. -----We will often indicate two vectors are perpendicular to each other by showing their dot product is zero.

    22. Example For the following vectors: Find the co-planar angle between the two vectors

    23. Example Find the proper value for the coefficient in So that the two vectors are perpendicular to each other.

    24. The dot product and orthonormal Coordinate Systems (CS) In this course we will almost exclusively use orthonormal CS What do we mean by orthonormal? We mean the basis vectors are orthogonal (perpendicular) and each have a magnitude of 1. Look at the results of application of the rectangular basis vector with the dot product: Magnitude = 1: And Orthogonal:

    25. Vector properties - Cross Product Another way to multiply two vectors together such that the resulting product is a vector For vectors The cross product is defined algebraically as The right-most equality shows the cross product expressed in terms of the determinant.

    26. Vector properties - Cross Product There also is a geometric definition: Where is the co-planar angle between and is a unit vector that is normal to the co-plane formed by vectors and . It can be found using the right hand rule. By the dependence on we can see the cross product is a measure of how perpendicular two vectors are to each other. -----We will often indicate two vectors are parallel to each other by showing their cross product is zero.

    27. Example The geostrophic wind equation can be expressed as where is the velocity vector, and and are vectors that we will learn about later in this course. Show What can you infer about the angle between and ?

    28. Vectors Definition revisited A common definition of a vector is an object with a magnitude and direction. A more useful definition is to look at how the above two properties are invariant to rotations or perspective of the observer.

    29. Vectors Definition revisited For a general 2-D vector The relationship between the coordinates of the vector in an un-rotated CS and a CS rotated degrees is

    30. Vectors Definition revisted Relationship of primed to unprimed coordinates:

    31. Exercise The magnitude of the vector in terms of unprimed coordinates is What is the magnitude in terms of primed coordinates?

    32. The previous exercise shows us that the vector magnitude is invariant to rotations. What about the dot product? It is also invariant to rotations as can be seen in the following proof. Vectors Definition revisited

    33. For two vectors and we can use the additive property to form a third vector. We can express the magnitude of as Rearranging for the dot product on the right side: We can see all terms on the right hand side of the above expression consists solely of vector magnitudes and thus is invariant to CS rotations. Consequently the left hand side must also be invariant to CS rotations therefore the dot product is invariant to CS rotations Vectors Definition revisited

    34. Finally from the geometric definition of the dot product We know that the left hand side is invariant to CS rotations so the right side must be also. We already proved that the magnitudes of the vectors are invariant so we can also include that the co-planar angle must also be invariant to CS rotations Vectors Definition revisited

    35. In summary we have observed the following about vectors 1. The magnitude of a vector is invariant under coordinate rotation. 2. The dot product of two vectors is invariant under coordinate rotation. 3. The angle between two vectors is invariant under coordinate rotation. This is important because it means that the physics of a problem will not change due to a change in perspective (change in CS). Vectors Definition revisited