1. Plot the point given in polar coordinates, and find its rectangular coordinates.
.
2. The rectangular coordinates of a point are (3, 4). Find two pairs of polar coordinates for the point, one with and the other with . Express in radians. (11)
3. Write the polar equation as an equation in rectangular coordinates. Identify the equation and graph it. (17)
4. Sketch the graph of the polar equation: . (23)
5. Write the complex number in the standard form . (30)
6. If and (36)
a) Find
b) Find
7. Write in the standard form (42)
8. Find all the complex fourth roots of . (50)
9. For the points and , the vector v is represented by the directed line segment . Write v in the form ai + bj and find (56)
10. Use the vectors v = and w to find (64) a) (64)
b) A unit vector in the opposite direction of w. (68)
11. Find the distance from to (74)
12. A vector v has initial point and terminal point . Write v
in the form v = ai + bj + ck. (76)
13. If v = 3i + j k and w = i + 2j k, a) Find v w. (83)
b) Find a unit vector orthogonal to both v and w. (86)
14. Find the dot product and the angle between v and w for
a) v = 3i j and w = i + j (88)
b) v = i j k and w = i + j k (94)
15. Determine whether v and w are parallel, orthogonal, or neither.
v = i j and w = 2i + j (96)
16. Decompose v into two vectors, one parallel to w and the other orthogonal to w.
v = i j and w = i + j (102)
17. Find the direction angles of the vector v = i j k (106) 18. Find the area of the parallelogram with vertices
(108)
19. An airplane has an airspeed of 500 km/hr in a northerly direction. The wind velocity is 60 km/hr in a southeasterly direction. Find the actual speed and direction of the plane relative to the ground. (112)
20. Find the work done by a force of 5 pounds acting in the direction 60o to the horizontal in
moving an object 20 feet from (0, 0) to (20, 0). (114)
21. A moving van with a gross weight of 8000 pounds is parked on a street with a 5o grade.
1. Plot the point given in polar coordinates, and find its rectangular coordinates.
.
2. The point (3, 4) lies in quadrant I.
Polar coordinates of the point (3, 4) are or
3.
The graph is a line with y-intercept (0, 2) and slope
4. Sketch the graph of the polar equation: . Check for symmetry:
Polar axis: Replace by . The result is
The graph is symmetric with respect to the polar axis.
The line : Replace by
The test fails
The pole: Replace r by . The test fails
Due to symmetry with respect to the polar axis, assign values to from 0 to .
0
5.
6. If and
a)
7.
8.
The four complex fourth roots of are:
9. and ,
v .
10. Use the vectors v = and w to find a)
b)
11. ,
12. .
v
13. If v = 3i + j k and w = i + 2j k,
a) v w
So, or
14. Find the dot product and the angle between v and w for
a) v = 3i j and w = i + j
v w
b) v = i j k and w = i + j k v w
15. Determine whether v and w are parallel, orthogonal, or neither.
v = i j and w = 2i + j
v w
Thus, the vectors are parallel
16. v = i j and w = i + j
The decomposition of v into 2 vectors and so that is parallel to w and is perpendicular to w is given by: and
17. Find the direction angles of the vector v = i j k
18.
u ; v
u v
Area = sq. units
19. Let = the velocity of the plane in still air, = the velocity of the wind, and
= the velocity of the plane relative to the ground.
;
The speed of the plane relative to the ground is:
km/hr
To find the direction, find the angle between and a convenient vector such as due north, j.
20.F
D
ft-lb
21. Split the force into the components going down the hill and perpendicular to the hill.
Fd
F Fp
Fd = F
Fp = F
