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Consequently a simple point normal equation for the

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Consequently, a simple point-normal equation for the plane is given by 2(x-4) - 3(y+5) - 3(z-6) = 0.
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TEST1/MAC2313 Page 4 of 5 _________________________________________________________________ 13. (5 pts.) Find the exact value of the acute angle θ of intersection of the two planes defined by the two equations x - 2y = -55 and 3y - 4z = 75. θ = cos -1 (( v w )/( v w )) = cos -1 (6/(125) 1/2 ) where v = <1,-2,0> and w = <0,3,-4>. Observe acutely the funny absolute value thingies. Why do we need them generally to get acute angles?? _________________________________________________________________ 14. (5 pts.) Write an equation for the plane which contains the line defined by <x,y,z> = <1,2,3> + t<3, -2, 1> and is perpendicular to the plane defined by x - 2y + z = 0. Since a normal vector v for the plane must be perpendicular to both the direction of the line, due to the line lying in the desired plane, and any non-zero normal vector for the given plane defined by the equation x - 2y + z = 0, we may build v by means of a cross product. Set v = <3,-2,1> × <1,-2,1> = <0,-2,-4>. A point-normal equation for the plane we want is now cheap thrills: -2(y - 2) - 4(z - 3) = 0 ... ugh. _________________________________________________________________ 15. (5 pts.) What is the radius of the sphere centered at (1, 0, 0) and tangent to the plane defined by x + 2y + z = 10? The radius,r , of the sphere will simply be the distance from the point (1,0,0) to the plane defined by the equation x - 2y + z - 10 = 0. 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??
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