102f05mt1 - B U Department of Mathematics Math 102 Calculus...

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Unformatted text preview: B U Department of Mathematics Math 102 Calculus II Fall 2005 First Midterm This archive is a property of Bo˘gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Let ~ a and ~ b be vectors. Show that (i) | ~ a · ~ b | ≤ | ~ a || ~ b | . If we have equality, what can you say about ~ a and ~ b ? Solution: By the definition of dot product we know that | ~a · ~ b | = | ~a || ~ b | cos θ = | ~a || ~ b || cos θ | Since 0 ≤ | cos θ | ≤ 1 | ~a · ~ b | ≤ | ~a || ~ b | If we have equality, | cos θ | = 1. Thus, θ = 0 or θ = π ....
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This note was uploaded on 04/30/2008 for the course C CMPE 150 taught by Professor Tuna during the Spring '08 term at Boğaziçi University.

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102f05mt1 - B U Department of Mathematics Math 102 Calculus...

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