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Math 206 Section A
Review for Test 1
The test is on Friday, October 8 during our class time. The test will cover sections 1.1-1.10,
- Let A = (3, − 3 , 2) and B = (2, 1 , 4).
(a) Find the distance between the points A and B.
(b) Find the midpoint of the segment from A to B.
(c) Write two parametrizations for the line passing through the points A and B. For each
parametrization, give the point when t = 0 and determine the value of t that gives the
point B on the line.
- Describe or sketch the relationship between the graphs of the following pairs of equations:
(a) z = −
x
2
2 and z =
x
2
2 − 4
(b) z = x
2 and z = (x − 1)
2
- Describe or sketch the set of points that satisfy the following equations:
(a) x
2
2
2
− 2 x + 4y + 10z + 21 = 0
(b) y = 3 − x
2 − z
2
(c) y
2
2 = 4
(d) x − 5 = 0
- Describe or sketch the set of points that satisfy the inequality: x
2
2
≤ z
2
, 0 ≤ z ≤ 2.
- Describe the curve in which the two surfaces x
2
2 = 5 and x
2
2
2 = 9 intersect.
(Assume z > 0). Write a paramterization for this curve.
- Find cylindrical coordinates of the points with the following rectangular coordinates:
(a) (− 1 , 0 , 2) (b) (− 1 ,
3 , 13) (c) (5, 6 , 3)
- Find spherical coordinates of the points with the following rectangular coordinates:
(a) (1, − 1 ,
- (b) (0,
- Convert the rectangular coordinate equation z
2
= 2x
2
2
to (a) cylindrical coordinates and
(b) spherical coordinates.
- Consider the two planes x + y = 1 and y + z = 1 that intersect in a straight line.
(a) Find the angle between the planes.
(b) Find a parametrization for the line of intersection.
- Suppose a spaceship is at position (100, 3 , 700) and is traveling with velocity (7, − 10 , 25).
Assume the units are in miles/sec for velocity. What is the position of the spaceship after 20
seconds? What is the speed of the spaceship?
- A ship is cruising due south at a rate of 15 knots (nautical miles per hour) with respect to still
water. However, there is also a current of 5
2 knots southeast. What is the total velocity of
the ship? If the ship is initially at the origin and a lobster pot is at position (20, −79), does
the ship hit or miss the lobster pot?
- Show that the line x = 5 − t, y = 2t − 7, z = t − 3 is contained in the plane having equation
2 x − y + 4z = 5.
- Does the line x = 5 − t, y = 2t − 3, z = 7t + 1 intersect the plane x − 3 y + z = 1? If so, where?
- A force
F =
i − 2
j acts on an object moving parallel to the vector ~a = 4
i +
j. What is the
force in the direction of motion?
- Let ~v = 2
i −
j +
k.
(a) Give a unit vector that points in the same direction as ~v.
(b) Give a vector of length 3 that points in the direction opposite to ~v.
- Find the area of the triangle determined by the vectors ~a =
i − 2
j + 6
k and
b = 4
i + 3
j −
k.
- Find a vector of length 2 that is orthogonal to both ~u = 2
i +
j − 3
k and ~v =
i +
k.
- Find an equation for the plane that is perpendicular to the line x = 3t−5, y = 7− 2 t, z = 8−t
and that passes through the point (1, − 1 , 2).
- Find an equation for the plane that contains the two lines: x = t + 2, y = 3t − 5, z = 5t + 1
and x = 5 − t, y = 3t − 10, z = 9 − 2 t.
- Find an equation for the line through the point (5, 0 , 6) that is perpendicular to the plane
2 x − 3 y + 5z = −1.
- Write parametric equations for the path of a particle that starts at the point (0, 0 , 3) and
travels clockwise along a circle of radius 3 in the xz-plane.
- Let T (x, y) = (3x + 4y, 7 x + 5y) be a linear transformation. Find the area of T (P ) where P
is a parallelogram of area 10.
The following problems are from the textbook.
- Pages 98-99: Problems 9, 31, 33
- Pages 131-132: Problems 3, 5, 7, 16(a, b, d, e), 17, 21
- Page 142: Problem 12(a, b)
- Pages 148-149: Problems 3, 11, 17
