Some Math 2A Problems to Consider in Studying for Chapter 9 Test

1.      True or False or Conditional.  Explain your answer. 

a.       Any two vectors determine a plane.

b.      Any two lines determine a plane.

c.       For a given point and a given plane there is a unique line through that point and perpendicular to the plane.

d.      The intersection of any two planes is a line.

e.       Any line in the xy plane will intersect a line in the xz plane.

 

2.      Find two vectors parallel to the plane  but not parallel to each other and compute the cross product of these vectors.

3.      Find the distance from the point (1,1,3) to the plane

4.      Show that the length of a vector is zero if and only if all its components are zero.

5.      Identify the surface whose equation is given as one of the following: a hyperbolic paraboloid, an elliptical cone, an elliptical paraboloid, an ellipsoid, a hyperboloid of one sheet or a hyperboloid of two sheets.

a.        

b.       

c.        

d.       

e.        

 

6.      The plane S passes through the point P(1, 2, 3) and contains the line
x = 3t, y = 1 + t, and z = 2  t. Find a vector normal to S.

7.      Which of the following statements is true for all three-dimensional vectors , if θ is the angle between ?  Note that none or all could be true.

a.        

b.       

c.        

d.       

8.      Find the torque at P if a 32 pound force is applied to the rigid body shown in the diagram at right. Note that this is a planar diagram.

9.      Describe the cross section that the given plane makes with the surface  

a.       x = 3

b.      y = x

c.        

 

10.  The Parallelogram Law states that   

a.       Give a geometric interpretation of this law.

b.      Prove the law.  The triangle inequality and/or Cauchy Schwarz inequality may be useful.

11.  Find the equation of the plane that contains the points (1, 2, 1), (2, 1, 0) and (3, 3, 1).

12.  Find the distance between the plane  and

13.  Parameterize the line segment from (6,4,1) to (3,2,5) as .

14.  Use vectors to prove that diagonals of a rhombus are perpendicular.

15.  Find the area of a parallelogram formed by vectors  and  if P(1,2,3), Q(5,4,2) and R(7,2,5).

16.  Show that for all , if x > 0 and  y > 0 , . 

17.  Write the equation in standard form:

18.  Suppose that Jack and Jill pull on a ropes attached to an object.  Jack pulls with a force of 450 N and Jill pulls with a force of 300 N.  The angle between the ropes is 30°.  With what direction and force should a third person pull so as to keep the object from moving?  Draw a diagram.

19.  Describe in words the surface whose equation is given.

a.        

b.       

c.      

20.  The cylindrical coordinates of a point are  

a.       What are the rectangular coordinates?

b.      What are the spherical coordinates?

21.  If we establish a spherical coordinate system centered at the center of Earth with the z-axis pointing through the north pole and the x-axis through the meridian line through Greenwich, England (the prime meridian) then latitude is  and longitude is .  Find the spherical coordinates of Palm Desert, CA (at latitude 116°, longitude 34°) and Santa Cruz, CA (at latitude 37°, longitude 122°) and find the great circle distance between these.

22.  Find the domain and range of the function

23.  Find the area of the largest square that can be contained in the unit cube.

24.  How can the triple product be used to determine whether or not three deifferent vectors are coplanar?  Give and example.

25.  Find the cross product of  and simplify.