Winter 2008 - Chan's Class - Exam 1

Winter 2008 - Chan's Class - Exam 1 - Jan 30, 2008 Math 20C...

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Jan 30, 2008 Math 20C Winter 2008 M IDTERM 1( A ) Solutions 1
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1. The following three space vectors are all parallel to the same plane which is this very sheet of paper: ±w ±v ±u where the vectors ±v and ±w are parallel to each other. For problems (i) – (vi) below, state whether the given object is: (a) the number zero, (b) the zero vector, (c) a nonzero vector perpendicular to the page, (d) a nonzero vector parallel to the page, (e) none of the above or undefined. No justification required. Each choice might appear more than once. (i) proj ±w ±v (d). The projection of any vector onto a vector ±w is always either a vector parallel to ±w , or the zero vector. Since ±v is neither zero nor perpendicular to ±w , its projection to ±w must be parallel to ±w , in particular to this page. (ii) ( ±w × ±u ) · ±v (a). ±w × ±u is perpendicular to the page because ±u and ±w are parallel to the page and not to each other. In particular, ±w × ±u is perpendicular to ±v . (iii) ( ±v · ±w ) ±u (d). ±v · ±w is a nonzero number, for the two vectors are not perpendicular to each other. So this is a nonzero scalar multiple of ±u , which is parallel to the page. (iv) ±u × ( ±u × ±v ) (d). ±u × ±v is perpenicular to the page, and ±u × ( ±u × ±v ) is perpendicular to ±u × ±v . The answer follows from the fact that any vector perpendicular to a normal vector
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Winter 2008 - Chan's Class - Exam 1 - Jan 30, 2008 Math 20C...

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