This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ......
View Full
Document
 '08
 LYMANA.PAGE
 Physics, mechanics

Click to edit the document details