VAM_13-18 - Tutorial V AM V ector Algebra and an...

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Tutorial VAM: Vector Algebra and an Introduction to Matrices summation convention, we simply drop the sum sign, and write a i ± ij = a j : Exercise 14. A ¢ B = a i b i = a i ( b j ± ij ) = a i b j ± ij : Exercise 15. Recall that the summation convention is in effect for all repeated indices. Thus, ² ijk a i b j ^x k involves sums over i; j and k: Since ² ijk vanishes if any two indices are equal, the subscripts onall threefactors in a i b j ^x k must be different. The Levi-Civitasymbol thencontributes a factor of 1 or ¡ 1 , depending on the order of the subscripts relative to (123) : Thus, a 1 b 2 ^x 3 will be multiplied by +1 and a 1 b 3 ^ x 2 will be multiplied by ¡ 1 ; according to Eqn. 30. The coefficients of all other terms are determined in the same way, giving the contributions to Eqn. 31. Finally, the coefficientof ^ x k in Eqn. 32 is the vector’s k th component. Exercise 16. A ¢ ( B £ C ) = a i ( B £ C ) i = a i ( ² jki b j c k ) = ² ijk a i b j c k : Note that we could
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