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Transcript
VECTORS "Worth seeing, yes; but not worth going to see" Samuel Johnson (said of the Giant's Causeway) In order to describe motion in more than one dimension it is convenient to introduce the concept of Vectors, which take into account both the magnitude and direction of certain quantities. A vector quantity must be specified by both a magnitude and direction. Examples are, velocity, force, displacement and acceleration . Contrast with scalar quantities which require only a magnitude. Examples of scalars are, mass, speed, time and distance. N.B. The magnitude of an object's velocity is its speed. Displacement is distance in a particular direction. Representation o Underline A o Bold A o Arrow on top o Graphical Addition and subtraction o Subtraction is the addition of a "negative" vector. "Negative" vector is opposite in direction with same magnitude as the original vector. o Graphical o Resolution. "Reverse" of addition. Any vector can be "resolved" into two (or more) components along perpendicular axes (x,y,z). o Vector addition (subtraction) via components 1. Resolve each vector into its x,y (z) components 2. Add (subtract) x and y components separately 3. Add x and y components to give resultant vector Specific representation of A o Magnitude: |A| = (Ax2 + Ay2 )1/2 o Direction: tan (theta) = Ay /Ax Since tan (theta) = tan (theta+180), sketching the vector will help define its direction exactly. Multiplication of vectors There are three forms of multiplication in common use o Multiplication by a scalar If k is a scalar and r is a vector, a = kr is a vector whose magnitude is given by |k||r| with direction either r or -r depending on the sign of k. o Scalar or dot product r and s are two arbitary vectors, r s | r || s | cos rs is the scalar product, where rs is the angle between r and s. r s rx s x ry s y rz s z In terms of components, Alternatively, the dot product can be thought of as the product of |r| and the component of s along r, | s | cos rs o Vector or cross product r x s uˆ | r || s | sin rs Where the unit vector u is at right angles to both r and s with a sense determined by the right hand rule. Using the unit vector notation for r and s the cross product can be evaluated directly or via the determinant method. i r s rx j k ry rz i sx sy sz ry rz sy sz j rx rz sx sz k rx ry sx sy i (ry s z rz s y ) j (rx s z rz s x ) k (rx s y “An Englishman, even if he is alone, forms an orderly queue of one” George Mikes – How to be an Alien (1946) Dr. C. L. Davis Physics Department University of Louisville email: [email protected]