week02 - Math 23 Sections 110-113 B Dodson Week 2 Homework...

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Math 23 Sections 110-113 B. Dodson Week 2 Homework: 12.2 vectors: unit, standard unit, notations 12.3 dot product: orthogonal, proj, comp 12.4 cross product: formula, properties Problem 12.2.25: Find a unit vector u that has the same direction as a = 8 i - j + 4 k. [variation/continuation: find a vector of length 4 in the opposite direction.] Solution: The length | 8 i - j + 4 k | = 64 + 1 + 16 = 81 = 9 , so the unit vector is u = 1 | a | a = 1 9 8 i - j + 4 k · = 8 9 i - 1 9 j + 4 9 k. Likewise, the vector with length 4 is - 4 9 8 i - j + 4 k · .
2 Problem 12.3.23bc: Determine whether the given vectors are othogonal, parallel or neither. (b) a = < 4 , 6 >, b = < - 3 , 2 > . (c) a = - i + 2 j + 5 k, b = 3 i + 4 j - k. Solution: (b) Recall that vectors are parallel if one is a scalar multiple of the other, a = cb, for a scalar c. This says < 4 , 6 > = c < - 3 , 2 > = < - 3 c, 2 c >, so 4 = - 3 c, and 6 = 2 c. The second equation gives c = 3 , but that’s not a solution of the first equation (with c = - 4 3 ), so there’s no solution c to the vector equation. Conclusion: the vectors in (b) are NOT parallel.