c3-t1-a(1)

# Using the appropriate formula we have r 112010 10 1 4

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Using the appropriate formula, we have r = (1)(1)+(2)(0)+(1)(0)-10 /(1 + 4 + 1) 1/2 = 9/(6) 1/2 . _________________________________________________________________ 16. (5 pts.) The equation z = 3 r 2 cos 2 ( θ ) is in cylindrical coordinates. Obtain an equivalent equation in terms of rectangular coordinates (x,y,z). Wake me up . .. z = 3x 2 .

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TEST1/MAC2313 Page 5 of 5 _________________________________________________________________ 17. (5 pts.) The point (5,-5 3,10) is in rectangular coordinates. Convert this to spherical coordinates ( ρ , θ , φ ). ( ρ , θ , φ ) = ( 10 (2) 1/2 , 5 π /3, π /4) . ρ 2 = 5 2 +(-5 3) 2 + 10 2 = 200 φ = cos -1 (1/(2) 1/2 ) = π /4 θ has its terminal side in the 4th quadrant, and its reference angle is θ r = tan -1 ( 3) = π /3. Consequently θ = 2 π - θ r = 5 π /3. _________________________________________________________________ 18. (5 pts.) Do the lines defined by the equations <x,y,z> = <3,1,2> + t<2,-1,-2> and <x,y,z> = <10,8,-5> + t<1,3,-1> intersect? If they do intersect, what is the point of intersection?? If the lines are to intersect, there must be numbers t 1 and t 2 so 3 + 2t 1 = 10 + t 2 , 1 - t 1 = 8 + 3t 2 , and 2 - 2t 1 = -5 - t 2 . [Here t 1 is the parameter for the putative point in terms of the first equation and t 2 is the parameter value for the second equation.] Solving this system yields t 1 = 2 and t 2 = -3. Using either t 1 or t 2 in the appropriate vector equation yields the point of intersection, (7,-1,-2). _________________________________________________________________ 19. (5 pts.) What is the area of the triangle in three space with vertices at P = (1, 0, 0), Q = (0, 2, 0), and R = (0 , 0, 3). Let v be the vector with initial point P and terminal point Q, and let w be the vector with initial point P and terminal point R. Then the area A of the triangle is given by A = v × w /2 = <-1,2,0> × <-1,0,3> /2 = <6,3,2> /2 = 7/2. _________________________________________________________________ 20. (5 pts.) Do the three 2-space sketches of the traces in each of the coordinate planes of the surface defined by y = 1 - x 2 - z 2 . Work below and label carefully. Then attempt to do a 3 - space sketch in the plane of the surface on the back of page 4.
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Using the appropriate formula we have r 112010 10 1 4 1 12...

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