mthsc206-fall-2010-notes-13.3

mthsc206-fall-2010-notes-13.3 - 13.3 The Dot Product...

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Unformatted text preview: 13.3 The Dot Product Algebraic Description Definition 13.3.1 Let u = < u 1 ,u 2 ,... ,u n > and v = < v 1 ,v 2 ,... ,v n > be two vectors in V n . The dot product u · v of u with v is the number given by u · v = u 1 v 1 + u 2 v 2 + · · · u n v n . NOTE: The dot product is also known as the scalar product or inner product. (Why do you think scalar product is an apt name?) Problem 13.3.1 Compute u · v for 1. u = v = < 2 , 3 , 6 > 2. u = 3 i + 4 j + 5 k and v =- i + 2 j- 3 k . 3. u = < 1 ,- 1 > and v = 2 i + j- k . Properties of the Dot Product: Let u , v , and w be vectors in V n , and suppose α is a real number. Then 1. u · u = || u || 2 2. u · v = v · u 3. u · ( v + w ) = u · v + u · w 4. ( α u ) · v = α ( u · v ) = u · ( α v ) 5. · u = 0. Proof of Property 2: Provide a proof of property 2 using the definition of dot product and the properties of real numbers. Geometric Description Definition 13.3.2 Define the angle between the vectors u and v to be the smaller of the two angles formed...
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This note was uploaded on 10/11/2010 for the course MTHSC 206 taught by Professor Chung during the Spring '07 term at Clemson.

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mthsc206-fall-2010-notes-13.3 - 13.3 The Dot Product...

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