VAM_49-59

VAM_49-59 - Tutorial V AM: V ector Algebra and an...

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Tutorial VAM: Vector Algebra andan Introduction to Matrices n 2 ¢ r =17 x +19 y +20 z = n 2 ¢ A =75 : The equations describing the line can thus be expresses parametrically as x = t; y =5 ¡ 3 t; z = ¡ 1 +2 t: The distance from the origin to the line is d = j ^ n 2 ¢ A j = 75 = p 17 2 +19 2 +20 2 = 5 p 42 = 14 : Exercise 49. Let the angle between A and ^x 1 be μ: Then, remembering that A 0 = A; a 0 1 = A 0 cos( μ + ' ) = A (cos μ cos ' ¡ sin μ sin ' ) = a 1 cos ' ¡ a 2 sin ': a 0 2 = A 0 sin( μ + ' ) = A (cos μ sin ' +sin μ cos ' ) = a 1 sin ' + a 2 cos ': These equations can be written in matrix form as in Eqn. 77. Exercise 50. R (2) (30 ± ) = μ cos(30 ± ) ¡ sin(30 ± ) sin(30 ± ) cos(30 ± ) = 1 2 μ p 3 ¡ 1 1 p 3 : Exercise 51. Let B also be rotated by R (2) ( ' ) : μ b 0 1 b 0 2 = μ cos ' ¡ sin ' sin ' cos ' ¶μ b 1 b 2 : Then, A 0 ¢ B 0 = a 0 1 b 0 1 + a 0 2 b 0 2 = ( a 1 cos ' ¡ a 2 sin ' )( b 1 cos ' ¡ b 2 sin ' ) +( a 1 sin ' + a 2 cos ' ) ( b 1 sin ' + b 2 cos ' ) = a
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VAM_49-59 - Tutorial V AM: V ector Algebra and an...

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