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HW13 - x What is the set of points described by this...

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HW 13 Algebra II Honors – Mr. Petach Please write your answers on a separate sheet! 1. Find the equations of the projecting planes of the lines containing the following pairs of points. Then, sketch one of the projecting planes and the line. (a) (4 , 0 , 4) and (0 , 2 , 2) (b) (0 , - 3 , 0) and (1 , 3 , 3) (c) (1 , 1 , 7) and (3 , 7 , 1) (d) (3 , 4 , 5) and (3 , 6 , 8) 2. Find the equations of the projecting planes of the following lines. (a) x = - 1 + 2 t y = 2 z = 2 - t (b) x = 1 - 2 t/ 3 y = - 1 + 2 t/ 3 z = 1 - t/ 3 (c) x = 1 + t y = 3 t z = 1 - t (d) x = 1 + 3 t y = 4 - 3 t z = - 2 + 7 t 3. Two planes are described by the linear equations x + y + 3 z = 9 and 2 x + y + 2 z = 10. (a) How would you describe the set of points contained in the intersection of the two planes? Is it a point? a line? a plane? (b) “Solve” the system of two equations by eliminating z . What is the set of points described by this solution? Is it a line? a plane? a sphere? (c) “Solve” the system of two equations by eliminating y . What is the set of points described by this solution? (d) “Solve” the system of two equations by eliminating
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Unformatted text preview: x . What is the set of points described by this solution? (e) The equations you found in parts (b), (c), and (d) are the projecting planes of the line of intersection of the two planes. To find the equation of the line, we need to use parametric equations. Parameterize x as x = t . Now, find the parametric equations for y and z by solving the following system of equations for y and z in terms of t : ( t + y + 3 z = 9 2 t + y + 2 z = 10 (f) In the previous part, you found the parametric equations of the line of intersection. Find two points on this line. Show that each point satisfies the equations of both planes. 1 4. Find the equations of the planes containing the following points: (a) (1, 1, -2), (0, -1, 1), and (-3, -2, 5) (b) (1, 1, 3), (0, 1, 1), and (-1, 0, 2) (c) (1, 1, 2), (4, 0, 0), and (-1, 1, 4) (d) (1, 0, 3), (1, 7, 3), and (-1, 1, 4) 2...
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