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Unformatted text preview: Tutorial VAM: Vector Algebra and an Introduction to Matrices 9. Solutions to Exercises and Problems Exercise 1. j A B j = AB j cos j AB since j cos j 1 : Exercise 2. j A + B j = p ( A + B ) ( A + B ) = p A 2 + B 2 + 2 A B p A 2 + B 2 + 2 AB = q ( A + B ) 2 = A + B: Exercise 3. Consider the sides of the triangle as vectors, oriented such that C = A B : Then, C 2 = ( A B ) ( A B ) = A 2 + B 2 2 A B = A 2 + B 2 2 AB cos C : Exercise 4. The magnitude of both sides is clearly the same. However, using the right-hand- rule on the left hand side (rotating B into A ) results in a direction that is exactly opposite to that obtained from the rotation of A into B : The negative sign denotes this reverse of direction. Exercise 5. Scalar products: ^ x ^x = ^y ^y = ^ z ^ z = 1; ^x ^y = ^ x ^z = ^y ^ z = 0 Vector products: ^x ^x = ; ^x ^ y = ^z ^ x ^ z = ^ y ^ y ^x = ^z ^y ^ y = ; ^y ^z = ^ x ^z ^x = ^y ^ z...
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- Spring '08