Trig/Math Anal
HW NO.
Name_______________________No_____
SECTIONS
(Brown Book)
12-1/12-2
12-3
12-3
12-4
12-4
12-5
12-6
12-9
Review
V-1
V-2
V-2B
V-3
V-3B
V-4
V-5
V-6
V-7
ASSIGNMENT
DUE
√
Practice Set A
Practice Set B #1-12
Practice Set G
Practice Set C
Practice Set K
Practice Set D
Practice Set E
Practice Set F #1-3
Practice Set F #4-17
California (Math Analysis) Standard(s):
1.0 Translate between polar and rectangular coordinates and interpret polar coordinates and vectors graphically.
7.0 Demonstrate an understanding of functions and equations defined parametrically and graph them.
Next Test Date:
Practice Set A: Vectors
1. Navigation: A ship travels 200 km west from port and then 240 km due south before it is
disabled. Illustrate this in a vector diagram. Use trigonometry to find the course that a rescue
ship must take from port in order to reach the disabled ship.
2. Aviation: On graph paper, make a diagram that illustrates the velocity of an airplane heading
east at 400 knots. Illustrate a wind velocity of 50 knots blowing toward the northeast. If the
airplane encounters this wind, illustrate its resultant velocity. Estimate the resultant speed and
direction of the airplane. (The direction is the angle the resultant makes with due north
measured clockwise from north.)
U
U
U
U
Plot points A and B. Give the component form of AB and find AB
b gb g
3. A 1,  2 , B 3,  2
b
gb g
4. A 3,  5 , B 5, 1
Polar coordinates of point P are given and O is the origin. Draw vector OP and give its
component form.
5. P 6, 72
6. P 2, 43
b g
b g
3, 1g
, vb
8, 4g
, and w  b
6,  2g
7. Let u  b
. Calculate each expression.
b. u  v
c. 3u  w
uv
Find the coordinates of the point P described.
a.
b g b g 21 of the way from A to B.
8. A 0, 0 , B 6, 3 ;
533568001 Page 1
d. 3u  w
b g b g 45 of the way from A to B.
9. A 7,  2 , B 2, 8 ;
10. Physics: Suppose that you pull a child in a wagon by pulling a rope that makes a 60 angle
with the ground.
a. If the pulling force F is 40 lb in the direction of the rope, give the horizontal and vertical
components of the force.
b. Which component, horizontal or vertical, moves the wagon along the ground?
c. If the rope made a 50 angle with the ground rather than a 60 angle, would the wagon move
more easily or less easily? Why?
Practice Set B: Vectors and Parametric Equations
Find vector and parametric equations for each specified line.
1. The line through (1, 5) with direction vector
2. The line through (1, 0) and (3, -4)
(2, -1)
3. The line through (-2, 3) and (5, 1)
4. The horizontal line through  , e
U
U
bg
b g
5. A point moves in the plane so that its position P x , y at time t is given by the equation
x, y  1,4  t 3,2
a. Graph the point’s position at the times t  0, 1, 2, 3, -1, - 2, and - 3
b. Find the velocity and speed of the moving point.
c. Find the parametric equations of the moving point.
6. Find the vector and parametric equations of the moving object with velocity (3, -1) and position
vector at time t  0 is (2, 3).
7. A line has vector equation x, y  3,2  t 2,4 . Give a pair of parametric equations and a
Cartesian equation of the line.
8. A line has parametric equations x  5  t and y  4  2t . Give a vector equation and a Cartesian
equation of the line.
9.
a. Describe the line having parametric equations x  2 and y  t .
b. Give a direction vector of the line.
c. What can you say about the slope of the line?
10.
a. A line has direction vector (2, 3). What is the slope of the line?
b. A line has direction vector (4, 6). What is the slope of the line?
c. Explain why the following lines are parallel:
d. Find a vector equation of the line through (7, 9) and parallel to these lines.
11. At time t, the position of an object moving with constant velocity is given by the parametric
equations x  2  3t and y  1  2t .
a. What are the velocity and speed of the object?
b. When and where does it cross the line x  y  2 .
12. An object moves with constant velocity so that its position at time t is x, y  1, 1  t 1, 1 .
b gb g b g
b gb g b g
b g
b gb g b g
When and where does the object cross the circle x  1  y 2  5 ? Illustrate with a sketch.
13. Without graphing, describe the curve with parametric equations x  r cos t and y  r sin t .
(Hint: What is the value of x 2  y 2 ?)
2
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14.
a. Before graphing, describe what you think the curve with parametric equations
x  2 cos t and y  5sin t looks like. Then graph the equations.
b. Find a Cartesian equation for the parametric equations given in part (a)
c. An ellipse has Cartesian equation 4 x 2  y 2  36 . What do you think the parametric
equations of the ellipse are? Check your answer by graphing.
Practice Set C: Parallel and Perpendicular Vectors; Dot Products
1. Find: a.  2, 3   4,  5
b.  3,  5   7, 4 
2. Find the value of a if the vectors (4, 6) and (a, 3) are
a. parallel
b. perpendicular
a. u  u
b. u
b g
4. If u  b
, verify that
5,  3g
and v  b
3, 7g
a. u  v  v  u
b. 2b
u  vgb
 2ug
v
5. Verify that u  b
v  wgb
 u  vgb
 u  wgif u  b
2, 5g
, vb
1, 3g
, and w  b
1, 2g
1, 3gand v  b
2, 1gis 45 .
6. Verify that the angle between u  b
3,  4gand v  b
3, 4g
7. Find the measure of the angle between u  b
.
3. If u  2, 3 , find:
2
8. Given A(1, 5), B(4, 6), and C(2, 8), find the measure of A .
2
9. Given P(0, 3), Q(2, 4), and R(3, 7), verify that cos P 
.
5
10.
a. Given A(1, -3), B(-1, 3) and C(6, 2), find cos C and sin C .
1
b. Use the formula Area  ab sin C to find the area of ABC
2
Practice Set D: Vectors in Three Dimensions
1. Find the length and midpoint of AB : A=(2, 5, -3) and B=(0, 3, 1)
2. Simplify
a.  3,8, 2  2  4, 1, 2 
b. 1, 8,6  5, 2,1
c.  3,5,1
3. Find the value of k if  2, k , 3 and  4, 2,6  are perpendicular.
4. Find an equation of the sphere with radius 7 and center (1, 5, 3). Show that point  7,7,6  is
on the sphere.
5. Find the center and radius of the sphere with equation x 2  y 2  z 2  2 x  4 y  6 z  11
6. Find the angle between (8, 6, 0) and (2, -1, 2) to the nearest tenth of a degree.
7. Let A  1,3, 4  , B   3, 1,0 , and C   3, 2,6  .
a. Show that AB and AC are perpendicular.
b. Find the area of right triangle ABC
8. Line L has vector equation  x, y, z    2,0,1  t  4, 1,1
a. Find three parametric equations of L.
b. Name two points on L.
c. Write a vector equation of the line containing (1, 2, 3) and parallel to L
9. Write vector and parametric equations for the line containing A  4, 2, 1 and B  6,3, 2  .
10. Where does the line  x, y, z    6, 5, 4  t 3, 2, 4  intersect
a. the xy-plane
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b. the yz-plane
c. the xz-plane
11. Describe the set of points S in the xy-plane that are also on the sphere whose equation
2
2
2
is  x  1   y  2    z  3  25 . Give an equation of S.
12. Show that the lines with equations  x, y, z    1,5,0  t 1, 2, 1 and
 x, y, z    0,1,3  s 1, 1,1 intersect.
Find the coordinates of their point of intersection.
Practice Set E: Vectors and Planes
1. Sketch the plane:
a. 2 x  3 y  6 z  12
b. 3x  y  2 z  6
2. Find a vector perpendicular to the plane whose equation is 3x  4 y  6 z  12
3. Find a Cartesian equation of the plane:
a. Vector (2, 3, 5) is perpendicular to the plane that contains point A(3, 1, 7)
b. Vector (1, -4, 2) is perpendicular to the plane that contains point A(3, 0, 2)
4. Consider the points A(2, 2, 2) and B(4, 6, 8)
a. Find a Cartesian equation of the plane that is perpendicular to AB at its midpoint M.
b. Show that the point P(2,0,8) satisfies your answer to part (a)
5. Find an equation of the plane tangent to the sphere  x  1   y  1   z  1  49 at the
2
2
2
point (7, -1, 4).
6. The plane z  3 intersects the sphere x 2  y 2  z 2  25 in a circle. Find the area of the circle.
7. To the nearest tenth of a degree, find the measure of the angle between the planes
2 x  2 y  z  3 and x  2 y  z  5 .
8. Are the planes 3x  4 y  2 z  5 and 2 x  y  z  3 perpendicular?
9. Which of the following planes are perpendicular and which are parallel?
M1 : 3x  2 y  z  6
M 2 : 6x  4 y  2z  8
M 3  4 x  2 y  8z  7
Practice Set F: Determinants and Vectors
1. Let u   4,0,1 , v   5, 1,0  , and w   3,1, 2 
a. Calculate v  u and u  v . Do your results agree with property 2?
b. Verify that u  v is perpendicular to u and to v .
c. Find the area of the parallelogram determined by u and v .
2. Let P(1,1, 0), Q(1, 0, 2), and R(2,1,1)
a. Find a vector perpendicular to the plane determined by P, Q, and R .
b. Find a Cartesian equation of the plane determined by P, Q, and R .
3. Angle  is between vectors u  (1, 2, 2) and v  (4,3, 0)
a. Find sin  by using property 3 of the cross product
b. Find cos by using the dot product property
c. Verify that sin 2   cos 2   1
Review Section:
4. An object is pulled due south by a force F1 of 5 N, and due east by a force F2 of 12 N. Find the
direction and magnitude of F3  F1  F2
5. Find the component form of AB and find AB if A(7, 2) and B(3, 4)
6. Given polar coordinates P(8,140) draw OP and find its component form.
533568001 Page 4
7. Find the coordinates of the point P, 14 of the way from A(1, 4) to B(5, 4)
8a. Find the velocity and speed of an object that moves with constant velocity so that its position
at time t is  x, y    2, 2  t 1, 4
8b. Find a pair of parametric equations of the path of the object.
8c. When and where does it intersect the parabola y  4 x 2  6 x ?
9. Find vector and parametric equations for the line through A(3,1) and B(4, 4)
10. Find a Cartesian equation for the vector equation  x, y    2, 3  t  5, 4 
11. Find the value of a if vectors (4, 2) and (a,8) are
a. parallel
b. perpendicular
12. If u   2,3 and v   4,1
a. find u  v
b. find the angle between u and v
13. Given A(4,3), B(5, 2), and C (8,1) , find mB .
14. Given the points A(4, 0, 5), and B( 6, 2, 7)
a. find the length of AB
b. find the midpoint of AB
15. Line L has equation  x, y, z    0, 1,6  t  4, 1, 3
a. Write a vector equation for the line through (2,1,7) parallel to L.
b. Where does L intersect the xz-plane?
16. Find an equation of the plane that is tangent to the sphere with equation
2
2
x 2   y  1   z  3  33 at the point (2,-3,-2).
17. A(0,1, 3), B(6,1,1) and C (4,5, 2) determine a plane.
a. find a vector perpendicular to the plane
b. find the area of triangle ABC
Practice Set G: Vectors and Parametric Equations
AB
Find AB and
:
1. A  3, 2 , B  4,3
2. A  4,1 B  7, 2 
Draw vector OP and give its component form:
3. P  4, 112
4. P  6,132
5. Let u   4,5 , v   4, 6 and w   2,8 . Calculate each expression.
a. 4u  w
b. u  3v
c. 5v
d. 4u  w
2
of the way from A(-2, 4) to B(7, -2)
3
3
7. Find the coordinates of the point P of the way from A(-6, 5) to B(2, 9)
4
Find vector and parametric equations for each specified line.
8. The line through (-1, 4) and (5, 8)
9. The vertical line through (2, -3)
6. Find the coordinates of the point P
533568001 Page 5
bg
10. A point moves in the plane so that its position P x , y at time t is given by the equation
 x, y   (4,3)  t (2, 5)
d. Graph
e. Find the velocity and speed of the moving point.
f. Find the parametric equations of the moving point.
11. A line has vector equation  x, y    1,3  t  4, 2 . Give a pair of parametric equations and a
Cartesian equation of the line.
12. A line has parametric equations
x  5  3t
. Write a vector and Cartesian equation for the
y  2  4t
line.
13. Line L has equation  x, y    4, 2  t  3,8 .
a. Write a vector equation for the line through (2,1) parallel to L.
b. Write a vector equation for the line through (4, -3) perpendicular to L.
14. The velocity of a plane heading west is 525 knots. It encounters a wind heading north east
with a velocity of 25 knots. Calculate the resultant speed and direction of the plane.
Practice Set K: Vectors and Parametric Equations
1. Find AB and AB : A  4, 3 , B  1,6
2. Draw vector OP and give its component
form: P  6, 232
3. Let u   3,6 , v   2, 1 and w   4,5 . Calculate each expression.
a. u  2w
b. 4u  2v
c. 3v
d. 4u  2v
e. u v
5
of the way from A(-4, 3) to B(2, -15)
6
Find vector and parametric equations for each specified line.
5. The line through (-5, 7) and (3, -2)
6. The horizontal line through (5, -3)
7. A point moves in the plane so that its position P x , y at time t is given by the equation
4. Find the coordinates of the point P
 x, y   (2, 5)  t (3, 1)
bg
a. Graph
b. Find the velocity and speed of the moving point.
8. A line has vector equation  x, y    4, 3  t  7,5 . Give a pair of parametric equations and a
Cartesian equation of the line.
9. An object moves with constant velocity so that its position at time t is  x, y    2,1  t  1,1 .
When and where does the object cross the line 3 x  4 y  15 ?
10. Find the value of a if the vectors (3a, 4) and (5, 7) are
a. parallel
b. perpendicular
u   4,7  and v   2,8
3u v 
11. If
find
u   5, 2
v   3,1
12. Find the measure of the angle between
and
.
13. Given A(2, 7), B(-3, 6), and C(5, -1), find the measure of ABC .
14. Given P(5, -2), Q(-3, 4), and R(6, 1), find the measure of QPR .
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x, y    5, 2  t  4,8
15. Line L has equation 
.
a. Write a vector equation for the line through (3,-5) parallel to L.
b. Write a vector equation for the line through (2, -3) perpendicular to L.
16. The velocity of a plane heading south is 340 knots. It encounters a wind heading south west
with a velocity of 65 knots. Calculate the resultant speed and direction of the plane.
ANSWERS
Practice Set A
1. 219.8
2. 436.8 knots; 85.4
6. (-1, -1.73)
7d. 10
10b. horizontal
5. (1.85, 5.71)
7c. (3, 1)
3. (2, 0); 2
7a. (-5, 5)
8. 3, 23
4. (-2, 6); 2 10
7b. (11, -3)
9. (3, 6)
bg
10c. easier
d
i
Practice Set B
x, ygb
 1,5g
 tb
2,1g
; x  1  2t , y  5  t
x, ygb
 1,0g
 tb
2,4g
; x  1  2t , y  4t
1. b
2. b
x, ygb
 2,3g
 tb
7,2g
; x  2  7t , y  3  2t
x, ygb
  , eg
 tb
1,0g
; x    t, y  e
3. b
4. b
5c. x  1  3t ; y  4  2t
5b. v  b
3,2g
; v  13
7. x  3  2t , y  2  4t ; 2 x  y  4
6. b
x, ygb
 2,3g
 tb
3,1g
; x  2  3t , y  3  t
9a. vertical line thru (2, 0)
x, ygb
 5,4g
 tb
1,2g
; 2 x  y  14
8. b
9b. (0, 1)
9c. slope is undefined 10a. 1.5
10b. 1.5
10c. slopes are equal
x, ygb
 7,9g
 tb
2,3g
10d. b
11b. t  1; b
5,3g
11a. v   3, 2  ; v  13
10a. 20, 20 3
bg
b g
12. t  1, 0,2 ; t  2, 3,1
13. circle; radius=r; center (0,0)
b g
14a. ellipse; vertices 0,5
14b. 25x 2  4 y 2  100
14c. x  3 cos t , y  6 sin t
Practice Set C
1a. -7
1b. 1
2. 2; -4.5
.
7. 106.3
8. 531
10a. cos C  0.6; sin C  0.8
Practice Set D
2a. (11, 6, 2)
2b. -5
1. 2 6 ; (1, 4, -1)
2
2
2
3. k=5
6. 70.5
5.  x  1   y  2    z  3  25
9.
 x, y, z   1, 2,3  t  4, 1,1
Vector:  x, y, z    4, 2, 1  t  2,1,3 ; Parametric:
10a. (3,-3,0)
10b. (0,-1,-4)
Practice Set E
2. (3, 4, 6)
5. 6 x  2 y  3z  56
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2c. 35
7b. 9
8b. ex: t  1,  2, 1, 2 ; t  2,  6, 2,3
8a. x  2  4t ; y  t ; z  1  t
8c.
3. 13; 13
10b. 20
10c.   32 ,0, 6
3a. 2 x  3 y  5 z  44
6. x 2  y 2  16; A  16
x  4  2t ; y  2  t ; z  1  3t
11.
 x  1   y  2 
2
2
 16
12. (-2, 3, 1)
3b. 1x  4 y  2 z  7
4a. 2 x  4 y  6 z  52
7. 47.1
9. M1 M 2 ; M1  M 3 ; M 2  M 3
Practice Set F
1c. 42
3b. 0.667
7. (2,2)
2a. (-1,4,1)
2b.  x  4 y  z  3
4. 112.6 , 13N
5. AB   4, 6  ; AB  2 13
8a.
v  (1, 4); v  17
3a. 0.745
6. (-6.14,5.14)
8c. t  32 :   12 , 4 ; t  5: 3,18
8b.
x  2  t ; y  2  4t
9.  x, y   (3,1)  t (7, 5); x  3  7t; y  1  5t
10. y  54 x  235
11b. 4
14a. 15.75
12a. -5
14b. (-1,-1,1)
12b. 109.7
13. 56.3
15a.  x, y, z    2,1,7   t  4, 1, 3
15b. (-4,0,9)
Practice Set G
1. AB   7,5  ;
16. 2 x  2 y  5 z  20
17a. (-16,46,-24)
11a. -16
17b. 27.1
2. AB  11, 3 ; AB  130
AB  74
3.  1.5,3.71
4.  4.01, 4.46 
5a.  18, 28
5b.  16, 23
5c.
5d. 33.29
6.  4, 0 
7. (0, 8)
1300
 x, y    1, 4  t  6, 4 ; x  1  6t; y  4  4t
9.  x, y    2, 3  t  0,1 ; x  2; y  3  t
10c. x  4  2t; y  3  5t
10b. velocity =  2, 5 ; speed = 29
11. x  1  4t ; y  3  2t ; 2 x  4 y  14
12.  x, y    5, 2  t 3, 4 ; 4 x  3 y  14
13a.  x, y    2,1  t  3,8
13b.  x, y    4, 3  t  8,3
8.
14. 272 ; 507.63
Practice Set K
1. AB   5,9  ; AB  106
3c.
45
3d.
2.  3.69, 4.73
932
3a.  11,16
3e. -12
 x, y    5,7  t 8, 9 ; x  5  8t; y  7  9t
6.  x, y    5, 3  t 1,0 ; x  5  t; y  3
8. x  4  7t ;
7b. velocity =  3, 1 ; speed = 10
20
9. when t  5 ; where  3, 6 
10a.
3b.  16, 26 
4. 1, 12
5.
11. 144
12. 176.6
15a.  x, y    3, 5  t  4,8
16. 388.69; 186.79
533568001 Page 8
y  3  5t ; 5 x  7 y  41
28
21
15
13. 52.5
14. 71.6
15b.  x, y    2, 3  t 8, 4 
10b. 
Given: u  ( x1 , y1 , z1 ) and v  ( x2 , y2 , z2 )
Vectors
uv  ( x2  x1 , y2  y1 , z2  z1 )
uv  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2
Vector Equation of a Line:
( x, y, z )  ( x0 , y0 , z0 )  t (a, b, c)
Parametric Equations of a Line:
x  x0  at; y  y0  bt; z  z0  ct
Dot Product: u  v  x1 x2  y1 y2  z1 z2
Properties:
1. u  v  v  u
Velocity: ( a, b, c)
3. k (u  v)  ku  v
Speed: velocity
2. If u  v  0, then u  v
4. u  (v  w)  u  v  u  w
Cartesian Equation of a Plane:
ax  by  cz  d , d  ax0  by0  cz0
where ( a, b, c) is the perpendicular direction vector
to the plane and ( x0 , y0 , z0 ) is a point on the plane.
Properties:
1. u  v is  to the plane containing u and v
Angle between two vectors:
u v
cos  
u v
2. u  v  (v  u )
i
j
k
Cross Product: u  v  x1
y1
y2
z1
z2
x2
3. sin  
uv
u v
4. u  (v  w)  (u  v)  (u  w)
5. If u  v  0, then u v
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