1. a) Determine the symmetric equations for the line throughP(5, 6, 10)and parallel to the line with equation= (6, 1, 1)+ t(2, 1, 3).

b) Determine two other points on this line

2.Determine parametric equations for the plane through the pointsA(2, 1, 1), B(0, 1, 3), andC(1, 3, 2).

3.Determine a vector equation for the plane that is parallel to thexy-plane and passes through the point(4, 1, 3).

4. Determine a scalar equation for the plane through the pointsM(1, 2, 3)andN(3 ,2, -1)that is perpendicular to the plane with equation3x + 2y + 6z +1 = 0.

5.Show that the line with parametric equationsx = 6 + 8t, y = 5 + t, z = 2 + 3tdoes not intersect the plane with equation2x y 5z 2 = 0.

6.Determine the intersection, if any, of the planes with equationsx + y - z + 12 =0and2x + 4y - 3z + 8 = 0.

7.Solve the following system of equations and give a geometrical interpretation of the result.

x + y + z = 6

2x + y 3z = -5

4x 5y + z = 3

8.Give a geometrical interpretation of the intersection of the planes with equations

x + y 3 = 0

y + z + 5 = 0

x + z + 2 = 0

9.Determine a scalar equation for the plane that passes through the point(2, 0, 1)and is perpendicular to the line of intersection of the planes

2x + y - z + 5 = 0andx + y + 2z + 7 = 0.

10.Explain why there are many different vector and parametric equations for a line.