Harold Washington College Math 209 Practice Problems Test 1 Part 2

Name: Date:

Solve the following problems.

(1) Let ~ u, ~ v be vectors. Prove the identity ||~ u~ v||2 +|~ u~ v|2 = ||~ u||2||~ v||2.

(2) Use the cross product of vectors to prove the Law of Sines for triangles.

(3) Find the distance from the plane 6z = 4y2x to the plane 13x + 6y = 9z. (4) Find the equation of the plane through the point (2,8,10) and parallel to the line x = 1 + t, y = 2t, z = 43t. (5) Is the line L1 through (4,6,1) and (2,0,3) parallel to the line L2 through (1,1,4) and (5,3,1)? Find a line L3 perpendicular to the line through (4,6,1) and (2,0,3) passing through the point (3,3,7). If the lines L2 and L3 intersect, nd the intersection point.

(6) If ~ A = i+j, ~ B = 2i3j+k, and ~ C = 4j3k, nd the angle between (~ A~ B)~ C and ~ A(~ B ~ C). (7) Find the area of the triangle having vertices (1,2,3), (2,1,1), and (1,2,3). (8) Two sides of a triangle are formed by the vectors ~ A = 3i+6j2k and ~ B = 4ij.Determine the angles of the triangle. Is this a right triangle?

(9) Find the projection of the vector 2i3j + 6k on the vector i + 2j + 2k. (10) Find the projection of the vector ~ u = 4i + 3j + k on the line passing throughthe points (2 ,3,1) and (2,4,3). 11) Does the plane 2x+4y +8z = 17 intersect the line x = 3+2t, y = t, z = 8t? If the answer is not positive, then nd the equation of the plane that contains the line x = 3 + 2t, y = t, z = 8t and is parallel to the plane 2x + 4y + 8z = 17.

(12) Determine whether the planes are parallel, perpendicular, or neither. If neither, nd the angle between them and the equation of the plane that perpendicular to the given planes. (a) x + 4y3z = 1, 3x + 6y + 7z = 0, (b) 2z = 4yz, 3x12y + 6z = 1, (c) x + y + z = 1, xy + z = 1, (d) x + 2y + 2z = 1, 2xy + 2z = 1. (13) Find the equation of the line that passes through (1,2,3) and is perpendicular to the plane that contains the points (0,0,2),(2,2,2) and (1,1,1).

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(14) (a) Find the distance from the point (1,2,4) to the plane 3x + 2y + 6z = 5. (b) What is the closest point in the plane 3x + 2y + 6z = 5 to the point (1,2,4)?

(15) Find the equation of the plane through the point (1,2,3) and containing the line x = 3t, y = 1 + t, z = 2t.

(16) Find the volume of the parallelepiped with adjacent edges P = (2,0,1), Q = (4,1,0), R = (3,1,1), and S = (2,2,2). What is the area of the faces of it?

(17) (a) Show the point (2,8,10) is not in the line x = 2 + t, y = 1 + 2t, z = 4 3t. (b) Find the equation of the plane through the point (2,8,10) and parallel to the line x = 2 + t, y = 1 + 2t, z = 43t.

(18) (a) Show the lines x = y = z and x + 1 =

y 2

=

z 3

are skew. (b) Find the

equation of the sphere that is tangent to the lines.

(19) Show that the distance between the parallel planes ax + by + cz + d1 = 0, and ax + by + cz + d2 = 0 is D = |d1 d2| a2 + b2 + c2. (20) Show that the lines with parametric equations x = 1+t, y = 1+6t, z = 2t, and x = 1 + 2t, y = 5 + 15t, and z = 2 + 6t are skew, and nd the distance between these lines.

(21) Find the equation of the planes that are parallel to the plane x + 2y 2 = 1 and two units away from it.

(22) Find the equation for the plane P1 consisting of all the points that are equidistant from the points (1,8,2) and (3,4,10). Find the equation of the plane that is 10 units away of P1.

(23) The intersection of planes x 2y + 4z = 2 and x + y 2z = 5 is a line L. Find the parametric and the symmetric equation of L. Find the plane that is perpendicular to the given planes.

(24) (a) Find the parametric equations for the line of intersection of the planes. (b) Find the angle between the planes (i) x + y + z = 1 and x + 2y + 2z = 1, (ii) 3x2y + z = 1 and 2x + y3z = 3. (25) The plane x + 3y z = 4 is intersected by the line L : x = 3 + 2t, y = 2t, z = t, nd the intersection point. Is the line L perpendicular to the plane? Find the equation of a line that is perpendicular to L and contains the intersecting point.

(26) Find the equation of the line L that is the intersection of the planes xz = 1 and y + 2z = 3, and the equation of the plane that contains L and is perpendicular to the plane xy2z = 1 (27) Let ~ u and ~ v two vectors. Show that ||~ u~ v||2 = ||~ u||2||~ v||2 (~ u~ v)2

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(28) Is the line of intersection of the planes x + 2y z = 2 and 3x + 2y + 2z = 7 parallel to the line x = 1+6t, y = 35t, z = 24t? Find the equation of the plane determined by these two lines.

(29) Prove that the lines

x1 =

y + 1 2

= z2

and

x2 =

y2 3

=

z4 2 intersect. Find the intersection point and the angle that these two lines make. Find the equation of a line that is perpendicular to these two lines.

(30) Find the equation of the plane through (3,3,1) that is perpendicular to the planes x + y = 2z and 2x + z = 10.

(31) Find the symmetric equations of the line through (2,4,5) that is parallel to the plane 3x + y2y = 5 and perpendicular to the line x + 8 2 = y5 3 = z1 1 . (32) Are the lines x = 2 + 2t,y = 1 + 4t,z = 2t and x + 2 20 = y1 40 = z2 10 parallel? Find the equation of the plane that contains L1 and L2.

(33) Do the lines

x1 4

=

y2 3

=

z4 2

and

x2 1

=

y1 1

=

z + 2 6 intersect? If the answer is yes, then nd the intersection point and the plane containing these lines.

(34) Show in at least four dierent ways that the points (2,1,3), (4,1,1), and (1,4,9) are collinear. (35) Are the points (1,3,2), (3,1,6), (5,2,0), and (3,6,4) in the same plane? Answer this question in two dierent ways.

(36) Find the volume of the parallelepiped determined by the vectors (1,5,2), (3,1,0), and (5,9,4). What you can conclude from your answer?