The point in question may now be obtained using the

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needed for the closest point. The point in question may now be obtained using the vector equation for the line. It is (8/6,5/6,5/6).

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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 - 3y = -5 and 2y - 4z = 7. θ = cos -1 (( v w )/( v w )) = cos -1 (6/(200) 1/2 ) = cos -1 (3/(5(2) 1/2 )) where v = <1,-3,0> and w = <0,2,-4>. Observe acutely the funny absolute value thingies. ______________________________________________________________________ 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. A normal vector n for the plane sought is since it must be perpendicular to a direction vector for the line and a normal vector of the given plane. [One may also obtain a suitable vector by solving an appropriate system of equations, a little two by three linear homogeneous thingy.] An equation for the plane is now cheap thrills: -2( y - 2) - 4( z - 3) = 0, or equivalently, y + 2 z - 8 = 0. ______________________________________________________________________ 15. (5 pts.) Obtain an equation for the plane tangent to the sphere defined by ( x 1) 2 ( y 2) 2 ( z 3) 2 9 at the point (2, -4, 5), which is actually on the sphere. Here all we need is a normal vector for the tangent plane. This can be obtained easily using the point of tangency and the center of the sphere at hand. An equation of the tangent plane: -( x - 2) + 2( y - (-4)) - 2( z - 5) = 0. You may obtain an equivalent standard form varmint if you wish. ______________________________________________________________________ 16. (5 pts.) The equation r 4cos( θ ) 10sin( θ ) is that of a cylinder in cylindrical coordinates. Obtain an equivalent equation in terms of rectangular coordinates (x,y,z). Provide a vector equation for the straight line that is the axis of symmetry. Multiply the given equation by r . Then doing the usual conversion and completing the square a couple of times yields ( x 2) 2 ( y 5) 2 29. A vector equation for the line of symmetry in 3-space is < x , y , z > <2, 5, t >, for t .
TEST1/MAC2313 Page 5 of 5 ______________________________________________________________________ 17. (5 pts.) The point (-3,-4,-5) is in rectangular coordinates. Convert this to spherical coordinates ( ρ , θ , φ ). [Inverse trig fun?] ______________________________________________________________________ 18. (5 pts.) Do the lines defined by the equations <x,y,z> = <0,1,2> + t<4,-2,2> and <x,y,z> = <1,1,-1> + t<1,-1,4> intersect? Justify your answer, for yes or no does not suffice.
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