14.1%20Ex%202-13

14.1%20Ex%202-13 - S E C T I O N 14.1 Vector-Valued...

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SECTION 14.1 Vector-Valued Functions (ET Section 13.1) 451 5. How do the paths r 1 ( t ) = h cos t , sin t i and r 2 ( t ) = h sin t , cos t i around the unit circle differ? SOLUTION The two paths describe the unit circle. However, as t increases from 0 to 2 π , the point on the path sin t i + cos t j moves in a clockwise direction, whereas the point on the path cos t i + sin t j moves in a counterclockwise direction. 6. Which three of the following vector-valued functions parametrize the same space curve? (a) ( 2 + cos t ) i + 9 j + ( 3 sin t ) k( b ) ( 2 + cos t ) i 9 j + ( 3 sin t ) k (c) ( 2 + cos 3 t ) i + 9 j + ( 3 sin 3 t ) d ) ( 2 cos t ) i + 9 j + ( 3 + sin t ) k (e) ( 2 + cos t ) i + 9 j + ( 3 + sin t ) k All the curves except for (b) lie in the vertical plane y = 9. We identify each one of the curves (a), (c), (d) and (e). (a) The parametric equations are: x =− 2 + cos t , y = 9 , z = 3 sin t Hence, ( x + 2 ) 2 + ( z 3 ) 2 = ( cos t ) 2 + ( sin t ) 2 = 1 This is the circle of radius 1 in the plane y = 9, centered at ( 2 , 9 , 3 ) . (c) The parametric equations are: x 2 + cos 3 t , y = 9 , z = 3 sin 3 t Hence, ( x + 2 ) 2 + ( z 3 ) 2 = ( cos 3 t ) 2 + ( sin 3 t ) 2 = 1 This is the circle of radius 1 in the plane y = 9, centered at ( 2 , 9 , 3 ) . (d) In this curve we have: x 2 cos t , y = 9 , z = 3 + sin t Hence, ( x + 2 ) 2 + ( z 3 ) 2 = ( cos t ) 2 + ( sin t ) 2 = 1 Again, the circle of radius 1 in the plane y = 9, centered at ( 2 , 9 , 3 ) . (e) In this parametrization we have: x = 2 + cos t , y = 9 , z = 3 + sin t Hence, ( x 2 ) 2 + ( z 3 ) 2 = ( cos t ) 2 + ( sin t ) 2 = 1 This is the circle of radius 1 in the plane y = 9, centered at ( 2 , 9 , 3 ) .

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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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14.1%20Ex%202-13 - S E C T I O N 14.1 Vector-Valued...

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