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Unformatted text preview: Physics 103H/105 Problem Set 1 Solutions Problem 1 (3pts) Let a and b are unit vectors in the xy plane making angles and with the xaxis respectively. i is the j i () x y a b unit vector in the x direction and j is the unit vector in the y direction. (a) From vector addition we can write a = a x i + a y j . Using trigonometry, a x = cos , a y = sin , since the x y a a x y i j a length of a is unity. Hence a = cos i + sin j , as required. If we repeat the same argument for b and replace by , we get b = cos i + sin j . To show that cos(  ) = cos cos + sin sin , we use the dot product. The definition of the dot product for two arbitrary vectors p and q is p q =  p  q  cos , (1) 1 where  p  denotes the magnitude of p (similarly for q ) and is the angle between p and q . We can calculate the dot product in two different ways. Firstly we note that  a  =  b  = 1 and the angle between the vectors is  . This gives us a b = cos(  ) . Secondly, a b = (cos i + sin j ) (cos i + sin j ) = cos cos i i + sin sin j j + cos sin i j + sin cos j i . Now, i and j are unit vectors and have magnitude 1, and i j = j i = 0, since i j , so if we equate the results from the two ways of calculating a b we get cos(  ) = cos cos + sin sin , as required. (N.B. it doesnt matter if is bigger than since cos(  ) = cos(  )). Take the unit vector a = cos i + sin j = (cos , sin ). We can rewrite this in the form of a 2 1 matrix (rows columns) a = cos sin . (b) Multiply R ( ) by a . R ( ) a = cos  sin sin cos cos sin = cos cos  sin sin sin cos + cos sin = cos( + ) sin( + ) = c . We have found that the matrix product of R ( ) and a is another column vector, c . From part a) we know that c is another unit vector in the xy plane, which makes angle + to the xaxis, so the 2 effect of R ( ) is to rotate a anticlockwise by an angle . Aside: Matrix Multiplication If matrix multiplication is still a bit unclear, one way to think of it is row 1 row 2 row 3 ......
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 '08
 LYMANA.PAGE
 Physics, mechanics, Dot Product, Acceleration, Sin, Cos, θ

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