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Homework 1A

# Homework 1A - • f 1 = 0 i 2 j 0 k • f 2 = 1 i 0 j 1 k...

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APPM 2350 HOMEWORK 01 FALL 2010 Suppose that you are given a vector A in R 3 defned by A = a 1 i + a 2 j + a 3 k where i , j , and k are the standard basis vectors. 1. What are the projections oF A onto the standard basis vectors i , j , and k ? How do you interpret this result? 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 di±erent 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 ; the 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? 3. You are still not satisfed with your new basis vectors! So you try the Following vectors:
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Unformatted text preview: • f 1 = 0 i + 2 j + 0 k ; • f 2 = 1 i + 0 j + 1 k ; • f 3 = 1 i + 1 j + 0 k . Determine iF f 1 , f 2 and f 3 are orthogonal to each other. IF they are not orthogonal to each other, then perForm the Following steps: • set u 1 = f 1 ; • set u 2 = f 2 − proj u 1 f 2 ; • set u 3 = f 3 − proj u 1 f 3 − proj u 2 f 3 . What can you say about the orthogonality oF u 1 , u 2 , and u 3 ? Normalize the vectors u 1 , u 2 , and u 3 and call the results e 1 , e 2 , and e 3 , respectively. (²YI: this orthogonalization process is a special case oF the Gram–Schmidt procedure and it plays an important role in many areas oF applied mathematics.) 4. ²inally, suppose a 1 = 4, a 2 = − 3, and a 3 = 7. Express A in terms oF your new basis vectors e 1 , e 2 , and e 3 . VeriFy that your result is equal to A when written in the standard basis....
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