Math 280 Past Exam I Questions
2)
x2  z2  1
x2  y2  z 2
Find an equation of the plane that contains the point (1,3,0) and the line x  1  3t , y  2  t , z  3
1) Identify and sketch each of the following surfaces: a)
b)
3) Find an equation of the plane passes through (1,1,2) and (2,1,1) and perpendicular to the plane
3x  y  z  8
4) Give an example of an equation for an elliptic paraboloid and sketch it by first describing its trace
with respect to z = 3.
5) Does the line
x  3  t , y  2  t , z  1  2t intersect the plane 4 x  2 y  3z  1 ?
6)
a. State the right-thumb rule.
b. Is
A  B  C equal to
A  (B  C)
? Justify your answer.
2x  y  z  1
x  1  t , y  3  t , z  1  2t
c. Find the distance from the point (1,0,2) to the plane
d. Find the distance from (1,0,2) to the line
e. Find the slope of the curve
r  1 2 cos
at


2
f. Write <1,2,3> as a sum of a vector parallel to <0,3,1> and a vector orthogonal to <0,3,1>.
g. Find an equation of the plane determined by (1,5,4), (3,1,6), and (4,2,1).
h. Find an equation of the yz-plane.
z 2  y  x2 1
i.
Identify and sketch the surface
j.
Find parametric equations for the portion of parabola
y  x2
from (1,1) to (3,9)
k. Find the distance from (1,2,3) to the x-axis.
l. Find the distance from (1,2,3) to the xz-plane
7) Find parametric equations of the line passing through (1,5,3) and perpendicular to 3x + 5y - z=6.
8) Find an equation of the plane that passes through (1,1,0) and contains the line of intersection of
the planes
9)
x  y  z  2, 3x  y  z  2 .
Find the equation of the plane containing the two intersecting lines:
L1 : x  1  t , y  2  t , z  1  t ,    t  
L2 : x  1  4s, y  1  2s, z  2  2s    s  
10)
B  3i  j  2k onto A  3i  k
Find the angle between B  3i  j  2k and A  3i  k .
Does ( A  B)  C make sense? Justify your answer.
a) Find the projection of
b)
c)
d) Find the Cartesian equation for
x  t 2 , y  t 4  1, t  0 and describe the path.`
e) Find a parameterization of the line segment joining (2,0,2) and (0,2,0).
f)
g)
A  B  0 for two nonzero vectors A and B, then they are parallel.
x2 y2

 1 traced clockwise.
Find parametric equations for
4
9
Show that if
11) Determine whether the line through (1,0,1) and (4,-2, 2) intersect the plane x + y – z = 0. If it does,
find the point of intersection.
12) A given object of mass 5kg is suspended by two cables shown below. Find the magnitude of the
tension in each cable.
x  3  t , y  4  t , z  5  2t and the plane 2 x  y  3z  4 are
13) Determine whether the line
perpendicular.
14) Let u
 2,1,0 , v  1,0,2  .
orthogonal to v.
15) Find the distance between
Find the vector component of the projection of u on v and the vector
2 x  y  3z  4
and
6x  3y  9z  1
16) Find parametric equations of the line of intersection of the planes
3x  2 y  z  1
and
3x  2 y  z  5
17) Find an equation of the plane of the plane that is perpendicular to 8x-2y+6z=1 and passes through
(-1, 2, 5) and (2, 1, 4)
18) (5 points each)
a) Find the area of the triangle with vertices (1,3,-2), (2,4,5) and (-3,-2,2).
b) Find the angle between the planes
19) Identify
z 2  4x 2  4 y 2  4
2 x  3 y  z  3
and
4x  5 y  z  1
and sketch the equation by first finding its traces.
20) Determine whether the two lines
x  2  t , y  1  t , z  4  7t
and
x  4  5s, y  2  2s, z  1  4s are skewed, parallel, or intersecting.
21)
a) Find a Cartesian equation for x
b) Simplify
 4 sin t , y  cos t and indicate the direction of motion.
(u  v)  (u  v )
22)
A) Show that the curvature of a circle of radius 10 is 1/10. b) Show that the curvature of a
straight line is 0
23) A projectile is fired from the top of a building 10 m high with an initial angle of angle 30 degrees. If
the projectile travels 90 m before hitting the ground, find its initial speed. You must first find a(t),
v(t), and r(t).
24) Reparametrize
r (t )  4 cos t , 4 sin t , 3t 
as a function of the arclength measured from t = 0. Then
use the DEFINTION to compute the tangent vector as a function of s.
25) Find the a) unit tangent vector and b) unit normal vector c) Osculating plane at t = 0 for the curve
r (t )  cos t  t sin t , sin t  t cos t , 3 
26) Find an equation of the circle of curvature for the equation
27) Consider
r (t )  e t i  e t j  2tk
y
1
at x=1.
x
A) Find the length of the curve from
t  0 to t  ln 3 B) Find the
maximum speed and the minimum speed.
28) Let
r (t ) be the position vector of a moving particle.
True/false: If the speed of the particle is
constant, then the velocity vector is perpendicular to the acceleration vector. Justify your answer.
29) Compute the a) curvature b) torsion for c)
30) Consider
3
2
r(t )  ti  4t j
aN
for
r (t )  cos 2ti  sin 2tj  tk
a) Find the length of the curve from (0,0) to (4,32) b) Find the normal
vector at t = 1. (hint: you may use the fact that the curve is concave up when t = 1)
31) A gun has initial speed of 200 ft/sec from the ground level. Find two angles of elevation that can
be used to hit the target 800 ft away. You must not use the formula given in the textbook without
proof.
32) A baseball is thrown from the stand 96 feet above the field at an angle of 30 degrees from the
horizontal. When and how far away will the ball strike the ground if the initial speed is 32 ft/sec?
33) Consider
r (t )  5t 2 , t  A) Find the point(s) where the curve has the largest curvature.
B) Find an
equation of the circle of curvature at t = 0.
34) Consider
r (t )  2 cos t ,2 sin t ,6 
a) Express the arclength from (2,0,6) as a function of t and reparametrize the as a function of s.
b) Find the arclength from
(2,0,6) to (0,2,6)
c) Find the torsion of the curve.
35) Let
r(t )  t 2 , t , 3t  1 
a) Find B (the binormal vector) at t = 0.
b) Find an equation of the osculating plane at t = 0.
36) Let
a)
r (t )  2 cos ti  2 sin tj  tk
Find N(t )
b) Find an equation of the normal plane when t = 0.
c) Represent the curve with arclength s as parameter starting (2,0,0)
37) Find the maximum curvature of
y  ln x
38) A gun has initial speed of 200 ft/sec from the ground level. Find two angles of elevation that can
be used to hit the target 1000 ft away. You may not use the formulas given in the textbook.\
39) An object of mass m travels along the parabola
y  x 2 with a constant speed of 5 units/sec.
Compute the acceleration of the object at (1,1).
40) A shell is to be fired from the ground level at an elevation angle of 30 degrees. What should the
muzzle speed be in order for the maximum height to be 3000 ft? You must not use the formulas in
the textbook unless you first prove the formulas.
41) Find an equation of the circle of curvature at x = 0 for y = cosx.
42) a) Find an equation of a line tangent to
the rectifying plane for
r(t )  ln( t 2  1), t , t 2 
r (t )  cos t , sin t , t  1 
when t = 1. B) Find an equation of
at t = 0.
y  ln x
at x = 1.
r(t )  3t , 2 , sin 2t , cos 2t 
at t = 0.
43) (7 points) Compute and sketch an equation of the circle of curvature to the curve
(hint: you may use the formula: At the base point <a,b>, center =
44) (7 points) Find the maximum and minimum curvature of the curve
45) Compute the principle normal vector and torsion for the curve
46) For
 a, b  
1

N
y  3x 2
r(t )  t , t 
2
a) i) Sketch the curve and ii) sketch
T(1) and N(1)
47)
(true/false)
a) If the vectors
u
and
v are parallel, then u  v  0
b) (true/false) the acceleration vector of a particle moving around a circle points toward the
center of the circle.
c) If the _____________________ of a curve is 0, then it is a straight line.
d) (true/false)For any vectors
e) (true/false) For any vectors
f)
u, v, w , (u  v )  w  u  ( v  w )
u, v , u  v  v  u
(true/false) The domain of the function
f ( x, y)  x 2  y 2 is a clopen set.
g) A set is closed if it contains all of its _______________ points.
h) Find the distance from (1,2,3) to the z-axis _____________________
i)
Find the unit vector in the direction of <1,2,3>_____________________
47) (3 points each)
a) Find all the boundary points of the domain of
f ( x, y)  ln( x  y) 2
b) Sketch any three level curves to the surface
f ( x, y)  x 2  y
c) Sketch the level surface for
f ( x, y, z )  x 2  y 2  z
at
f ( x, y , z )  1
48) (5 points)
a) Simplify
(u  v)  (u  v) .
Justify each step carefully
b) Find the arclength function of
c) Determine whether the line
r (t )  sin 3t , cos 3t ,4t  starting at t=1.
L : x  1  t , y  3  2t , z  t
and the plane 3x+2y-5z=5 intersect or
not. If they do, find the point of intersection.
d) Find the distance from (1,4,3) to the plane
3x  y  z  1 .