Homework 1 Solutions

Homework 1 Solutions - WEEK ONE HW SOLUTIONS Let A = a1 i...

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WEEK ONE HW SOLUTIONS Let A = a 1 i + a 2 j + a 3 k And remember the formula for projection: proj B A = ° A · B B · B ± B 1. What are the projections of A onto the standard basis vectors i , j , and k ? How do you interpret this result? Answer. proj i A = ° A · i i · i ± i =( A · i ) i =( a 1 + 0 + 0) i = a 1 i Similarly proj j A = a 2 j and proj k A = a 3 k The projection of any vector onto the standard basis is the standard decomposition into the distance along the x-axis, y-axis and z-axis. 2. You have decided that you are more of a cylindrical person you think outside the box. In other words, you want to express A in terms of a diFerent set of basis vectors and have decided to use the following vectors: a unit vector e r that points in the direction a 1 i + a 2 j +0 k ; a unit vector e θ that points in the direction a 2 i + a 1 j +0 k ; a unit vector e z = k . Determine expressions for e r and e θ in terms of a 1 , a 2 , and a 3 , and the standard basis vectors. Remember that e r , e θ and e z must be unit vectors. What can you say about the orthogonality of these new basis vectors? Write A in terms of the three new basis vectors (Hint: think about projecting A onto the new basis vectors.) How do you interpret the projection of A onto each of these three new unit vectors?
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Homework 1 Solutions - WEEK ONE HW SOLUTIONS Let A = a1 i...

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