MIT16_07F09_Lec03

MIT16_07F09_Lec03 - S Widnall 16.07 Dynamics Fall 2009...

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S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors . Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = | A | . In handwritten text, vectors are often expressed using the arrow, or underbar notation, e.g. A , A . Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α , the result is a vector B = α A , which has magnitude B = | α | A . The vector B , is parallel to A and points in the same direction if α > 0. For α < 0, the vector B is parallel to A but points in the opposite direction (antiparallel). If we multiply an arbitrary vector, A , by the inverse of its magnitude, (1 /A ), we obtain a unit vector which ˆ is parallel to A . There exist several common notations to denote a unit vector, e.g. A , e A , etc. Thus, we have that A ˆ = A /A = A / | | , and A = A ˆ | A | = 1. A A , ˆ 1
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Vector addition Vector addition has a very simple geometrical interpretation. To add vector B to vector A , we simply place the tail of B at the head of A . The sum is a vector C from the tail of A to the head of B . Thus, we write C = A + B . The same result is obtained if the roles of A are reversed B . That is, C = A + B = B + A . This commutative property is illustrated below with the parallelogram construction. Since the result of adding two vectors is also a vector, we can consider the sum of multiple vectors. It can easily be verified that vector sum has the property of association, that is, ( A + B ) + C = A + ( B + C ) . Vector subtraction Since A B = A + ( B ), in order to subtract B from A , we simply multiply B by 1 and then add. Scalar product (“Dot” product) This product involves two vectors and results in a scalar quantity. The scalar product between two vectors A and B , is denoted by A B , and is defined as · A B = AB cos θ . · Here θ , is the angle between the vectors A and B when they are drawn with a common origin.
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