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ps1_soln

# ps1_soln - Physics 103H/105 Problem Set 1 Solutions Problem...

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Physics 103H/105 Problem Set 1 Solutions Problem 1 (3pts) Let ˆ a and ˆ b are unit vectors in the x-y plane making angles θ and φ with the x-axis respectively. ˆ i is the j i (θ-φ) x y a b θ φ unit vector in the x direction and ˆ j is the unit vector in the y direction. (a) From vector addition we can write ˆ a = a x ˆ i + a y ˆ j . Using trigonometry, a x = cos θ , a y = sin θ , since the x y a a x y θ i j a length of ˆ a is unity. Hence ˆ a = cos θ ˆ i + sin θ ˆ j , as required. If we repeat the same argument for ˆ b and replace θ by φ , we get ˆ b = cos φ ˆ i + sin φ ˆ j . To show that cos( θ - φ ) = cos θ cos φ + sin θ sin φ , we use the dot product. The definition of the dot product for two arbitrary vectors p and q is p · q = | p || q | cos α, (1) 1

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where | p | denotes the magnitude of p (similarly for q ) and α is the angle between p and q . We can calculate the dot product in two different ways. Firstly we note that | ˆ a | = | ˆ b | = 1 and the angle between the vectors is θ - φ . This gives us ˆ a · ˆ b = cos( θ - φ ) . Secondly, ˆ a · ˆ b = (cos θ ˆ i + sin θ ˆ j ) · (cos φ ˆ i + sin φ ˆ j ) = cos θ cos φ ˆ i · ˆ i + sin θ sin φ ˆ j · ˆ j + cos θ sin φ ˆ i · ˆ j + sin θ cos φ ˆ j · ˆ i . Now, ˆ i and ˆ j are unit vectors and have magnitude 1, and ˆ i · ˆ j = ˆ j · ˆ i = 0, since ˆ i ˆ j , so if we equate the results from the two ways of calculating ˆ a · ˆ b we get cos( θ - φ ) = cos θ cos φ + sin θ sin φ, as required. (N.B. it doesn’t matter if φ is bigger than θ since cos( θ - φ ) = cos( φ - θ )). Take the unit vector ˆ a = cos θ ˆ i + sin θ ˆ j = (cos θ, sin θ ). We can rewrite this in the form of a 2 × 1 matrix (rows × columns) ˆ a = cos θ sin θ . (b) Multiply R ( α ) by ˆ a . R ( α ) ˆ a = cos α - sin α sin α cos α cos θ sin θ = cos α cos θ - sin α sin θ sin α cos θ + cos α sin θ = cos( α + θ ) sin( α + θ ) = ˆ c . We have found that the matrix product of R ( α ) and ˆ a is another column vector, ˆ c . From part a) we know that ˆ c is another unit vector in the x-y plane, which makes angle α + θ to the x-axis, so the 2
effect of R ( α ) is to rotate ˆ a anti-clockwise by an angle α . Aside: Matrix Multiplication If matrix multiplication is still a bit unclear, one way to think of it is row 1 row 2 row 3 . . . row i . . . × ˆ column 1 column 2 column 3 . . . column j . . . ! . The entry in the i th row and j th column of the product matrix is the sum of the 1 st element in the i th row by the first element in the j th column with the product of the second elements in each respectively and so on. This means you can only take a matrix product between two matrices when the first has the same number of rows as the second has columns.

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ps1_soln - Physics 103H/105 Problem Set 1 Solutions Problem...

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