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vectorproblems_solutions

# vectorproblems_solutions - Problem 1 Prove that P = cos1a x...

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Problem 1 : Prove that y x a a P 1 1 sin cos θ θ + = and y x a a Q 2 2 sin cos θ θ + = are unit vectors in the xy -plane respectively making angles θ 1 and θ 2 with the x -axis. By means of dot product, obtain the formula for ( ) 1 2 cos θ θ . By similarly formulating P and Q , obtain the formula for ( ) 1 2 cos θ θ + . Solution: 1 sin cos 1 2 1 2 2 = + = θ θ P . Similarly, 1 2 = Q . So P and Q are unit vectors. The dot product gives us 2 1 2 1 sin sin cos cos θ θ θ θ + = Q P . We also know that ( 1 2 cos θ θ = Q P Q P ) where ( ) 1 2 θ θ is the angle between vectors P and Q . So, ( ) 2 1 2 1 1 2 sin sin cos cos cos θ θ θ θ θ θ + = . Let ( ) ( ) y x y x a a a a P 1 1 1 1 1 sin cos sin cos θ θ θ θ = + = and y x a a Q Q 2 2 1 sin cos θ θ + = = . Then by the dot product, we have 2 1 2 1 1 1 sin sin cos cos θ θ θ θ = Q P and ( ) ( ) ( ) 1 2 1 2 1 1 1 1 cos cos θ θ θ θ + = = Q P Q P . So, ( ) 2 1 2 1 1 2 sin sin cos cos cos θ θ θ θ θ θ = + . Problem 2 : Decompose vector z y x a a a 4 5 10 + into vectors parallel and perpendicular to vector y x a a + 3 . Solution: A = A - A || A B A || = ( A a B ) a B Let z y x a a a A 4 5 10 + = , y x a a B + = 3 and a B the unit vector in the direction of B . y x B a a B B a 10 1 10 3 + = = . ( ) y x B B a a a a A A 5 . 2 5 . 7 || + = = z y x a a a A A A 4 5 . 7 5 . 2 || + = = Problem 3 : E and F are vector fields given by z y x yz x a a a E + + = 2 , and . Determine: z y x xyz y xy a a a F + = 2 (a) | E | at (1, 2, 3) (b) the component of E along F at (1, 2, 3) (c) a vector perpendicular to both E and F at (0, 1, -3) whose magnitude is unity.

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