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1. a) Determine the symmetric equations for the line through P(3, 3, 5) and parallel to the line with equation F = (6, 1, 1) + t(-2, 1, 3). b) Determine two other points on this line. 2. Find the value of k so that the lines X - 3 _ y+6 2+3 3k +1 2k and 2+7 _ y+8_z+9 2 3 -2k - 3 are perpendicular. 3. Determine parametric equations for the plane through the points A(2, 2, 1), B(0, 1, 3), and C(3, 3, -2). 4. Determine a vector equation for the plane that is parallel to the xz -plane and passes through the point (4, 1, 3). 5. Determine a scalar equation for the plane through the points M(1, 2, 3) and N(3 ,2, -1) that is perpendicular to the plane with equation 3x + 2y + 6z + 1 = 0. 6. Show that the line with parametric equations x = 6 + 8t, y = -5 + t, z = 2 + 3t does not intersect the plane with equation 2x - y - 5z - 2 = 0. 7. Determine the intersection, if any, of the planes with equations x + y - z + 12 =0 and 2x + 4y - 3z + 8 = 0. 8. Solve the following system of equations and give a geometrical interpretation of the result. xty+z=6 2x + y - 3z = -5 4x - 5y + z =-3 9. Give a geometrical interpretation of the intersection of the planes with equations xty- 3=0 y+z+5=0 x+z+2=0 10. Determine a scalar equation for the plane that passes through the point (3, 1, -1) and is perpendicular to the line of intersection of the planes 2x+ y- z + 5 = 0 and x + y + 2z + 7 = 0. 11. Explain why there are many different vector and parametric equations for a line