HM6-soln - Solution for HW6 November 5, 2011 7.52 ~ A + ~ B...

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Unformatted text preview: Solution for HW6 November 5, 2011 7.52 ~ A + ~ B + ~ C = (2 + 1 + 3) ~ i + (- 1 + 1- 2) ~ j + (1 + 2 + 4) ~ j = 6 ~ i- 2 ~ j + 7 ~ k. And the unit vector is ~ A + ~ B + ~ C ~ A + ~ B + ~ C = 6 ~ i- 2 ~ j + 7 ~ k 6 ~ i- 2 ~ j + 7 ~ k = 6 ~ i- 2 ~ j + 7 ~ k 89 . 7.56 proj ~ B ( ~ A + ~ C ) = ( ~ A + ~ C ) ~ B ~ B = (2 + 3 , 1- 4 , 1 + 2) (1 ,- 2 , 2) p 1 2 + (- 2) 2 + 2 2 = 5 + 6 + 6 9 = 17 3 . 1 7.58 The rhombus has the property that ~ AB = ~ DC, ~ AB = ~ AD . The two diagonals can be expressed as ~ AB + ~ AD and ~ AB- ~ AD . Therefore ( ~ AB + ~ AD ) ( ~ AB- ~ AD ) = ~ AB ~ AB- ~ AD ~ AD = ~ AB 2- ~ AD 2 = 0 shows that the two diagonals are perpendicular to each other. 7.62 Not necessarily true. When ~ A and ~ B- ~ C are parallel we also have ~ A ~ B = ~ A ~ C . 7.68 Denote the given four vetices as A,B,C,D. V olumn = 1 6 ~ AB ( ~ AC ~ AD ) 2 = 1 6 1- 2- 1- 1 2- 1- 2 1- 1- 1- 1 1- 2- 2- 1 1- 1 = 1 6 8 = 4 3 . 7.69 First we have ( ~ A ~ B ) ( ~ C ~ D ) = ~ D [( ~ A ~ B ) ~ C ] = ~ D [( ~ A ~ C ) ~ B- ( ~ B ~ C ) ~ A ] = ( ~ A ~ C )( ~ D ~ B )- ( ~ B ~ C )( ~ D ~ A ) ....
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This note was uploaded on 02/28/2012 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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HM6-soln - Solution for HW6 November 5, 2011 7.52 ~ A + ~ B...

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