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In problems 1 through 3, let A = (1,1,1), B = (5,3,2), C=(2,5,3), and D=(3,2,5).

1.  Consider the parallelepiped with vertices at A, B, C, and D, such that B, C, and D are all adjacent to A (That is, there is an edge between A and B, between A and C, and between A and D). Find the coordinates of the vertex in the parallelepiped opposite to A.

a.  (6, 7, 4)

b.  (8, 8, 8)

c.  (7, 4, 6)

d.  (3, 5, 6)

e.  (6, 7, 5)

2.  Find the volume of the parallelepiped mentioned in number 5.

a.  7

b.  49/3

c.  49/2

d.  49

e.  56

3.  Find the volume of the tetrahedron with vertices at A, B, C, and D.

a.  49/6

b.  49/3

c.  49/2

d.  49

e.  56

4.  Find an equation of the plane through the points (1, 1, 1), (2, 2, 0), and (3, 0, 1). Example 3 in the section 11.5 may help.

a.  x - y = 0

b.  x + 2y + 3z = 6

c.  x + y - z = 1

d.  2x - y = 1

e.  x - y + z = 0

5.  Find parametric equations for the line through the point (1, 2, 3) parallel to the y-axis.

a.  x = t, y = 2t, z = 3t

b.  x = 1 + t, y = 2 + t, z = 3 + t

c.  x = 1 + t, y = 2 + 2t, z = 3 + 3t

d.  x = 1, y = t, z = 3

e.  x = 1 + t, y = 2 + t, z = 3

6.  Consider the line given by x = 3 - t, y = -2 + 3t, z = 2 + t, and the line given by x = 1 + s, y = 3 - s, z = 2 + 3s. Do these two lines intersect? If so, where do they intersect?

(Hint: Since x = 3 - t in the first line, and x = 1 + s in the second line, set these two equal like so: 3 - t = 1 + s. Now do the same for the "y"s and "z"s. You should now have 3 equations in the two variables s and t. See if those three equations can be solved for s and t. If they can't, the lines don't intersect. If they can, plug back in to find x, y, and z.)

a.  The lines don't intersect.

b.  The lines intersect at (-1, 3, 1).

c.  The lines intersect at (2, 1, 3).

d.  The lines intersect at (1.5, 2.5, 3.5)

e.  The lines intersect at the origin.

7.  Given a line in space and a point not on the line, does there always exist a plane containing both the line and the point?

a.  Yes

b.  No

8.  In \(R^3\) the equation \(y = 2x + 1\) represents

a.  A straight line with slope 2.

b.  A vertical plane touching the y-axis at y = 1.

c.  A horizontal plane touching the z-axis at z = 1.

d.  A plane whose normal vector is parallel to the x-axis.

e.  A plane whose normal vector is parallel to the y-axis.

9.  In \(R^3\) the equation \(z = 2\) represents

a.  A vertical plane perpendicular to the x-axis.

b.  A vertical plane perpendicular to the y-axis.

c.  A horizontal plane

10.  The equation 2x - y + 3z = 9 represents a plane

a.  With normal vector \(\langle 2, 1, 3\rangle\) through point \((9, 0, 0)\)

b.  With normal vector \(\langle 2, 1, 3\rangle\) through point \((0, 0, 3)\)

c.  With normal vector \(\langle 2, -1, 3\rangle\) through point \((9, 0, 0)\)

d.  With normal vector \(\langle 2, -1, 3\rangle\) through point \((0, 0, 3)\)

e.  With normal vector \(\langle 2, -1, 3\rangle\) through point \((0, 0, 9)\)

11.  The planes \(x + 2y + 3z = 6\) and \(2x + 4y + 6z = 6\) are

a.  the same plane.

b.  intersecting planes.

c.  Parallel planes.

d.  Non-parallel planes that don't intersect