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ASE 366K - Vector Calculus Primer

# ASE 366K - Vector Calculus Primer - 4 20 A ppendix Vectors...

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420 Appendix. Vectors and Vector Calculus In a coordinate system with orthonormal vectors (i, j, k), we can find the coordinates for a x b in terms of the coordinates of a and the coordinates of b. ixj : k jxk : i kxi j Note: the vectors i, j, and k are oriented in such a way that they form the basis of a right-handed coordinate system. Using the above formulae, and the anti-commutative law, one obtains the cartesian coordinate expression for the cross product: a x b (a,i -f aoj -t a"k) x (b"i -F bai + b"k) (oob" - boa")i I (a"b* b"a,)i I (a*b, b,ao)k. Briefly, & X b (oob, boar,a"b* b"a*,a*bs b"ay) . An important formula which we shall use several times, and which can be verified by direct (and repeated) use of the coordinate expression for the vector product, is the so called "bac-cab" formula: a x (b x c) b(a' c) c(a'b) The scalar and vector products give rise to a number of similar identities that can be verified by checking the coordinate expressions of both sides of the equation. Below we list just a couple of these: a' (b x c) b' (c x a) c' (a x b), (a x b) ' (c x d) (a. c)(b.d) (".d)(b'c). Vector Calculus Consider a vector r which is the position vector for a particle P (see the figure below). Suppose
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ASE 366K - Vector Calculus Primer - 4 20 A ppendix Vectors...

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