Polars and Vector Review Sheet
Pre-Calc. for AP Prep.
Name: ______________
Date: ______________
No Calculators
Convert to rectangular coordinates.
1)
5 

 3,

4 

2)
Convert to polar coordinates.


 6,  
2

3)
 2, 2 3 
4)
(-6, 0)
Plot each point in polar coordinates. Then find another representation  r ,  of this point which
meet the conditions in parts a to c.
5)
 3 
 4,

 4 
6)
y
5 

 3,

6 

7)
y
y
x

 

5 

 2, 

3 

x

 

x

 
a. r > 0, 2 < θ < 4
a. r > 0, 2 < θ < 4
a. r > 0, 2 < θ < 4
b. r < 0, 0 < θ < 2
b. r < 0, 0 < θ < 2
b. r < 0, 0 < θ < 2
c. r > 0, - 2 < θ < 0
c. r > 0, - 2 < θ < 0
c. r > 0, - 2 < θ < 0

Convert each rectangular equation into a polar equation.
8)
5x – y = 7
9)
y = -7
10)
Convert each polar equation to a rectangular equation.
(x + 1)2 + y2 = 1

11)
r=6
12)

13)
r = -3cscθ
14)
r = 4sinθsec2θ
3
Graph the polar equations.
15)
θ
0

6

4

3

2
2
3
3
4
5
6

r = -4sinθ
r
θ
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
r
y
x




16)
θ
0
r = 2sin3θ
r
θ
r
7
6
5
4
4
3
3
2
5
3
7
4
11
6

6

4

3

2
2
3
3
4
5
6
y
x




2

Let v be the vector from initial point P1 to terminal point P2. Write v in terms of i and j.
17)
P1 = (6, 4); P2 = (-5, -4)
18)
P1 = (2, 3); P2 = (-6, 3)
Find the specified vector or scalar.
19)
u = -7i - 3j, v = -5i + 7j; Find u + v.
20)
u = -9i - 2j, v = 5i + 7j; Find u - v.
21)
v = 8i + 2j; Find 3v.
22)
v = -7i + 2j; Find 9v .
23)
u = 4i - 8j and v = 14i - 13j; Find u · v.
24)
u = 6i + 10j, v = -2i - 8j, w = -4i - 2j;
Find u · (v + w).
Find the unit vector that has the same direction as the vector v.
25)
v = 2i
26)
v = 5i + 12j
27)
Write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are
||v|| = 10, θ = 120°
28)
Determine if orthogonal: v = 3i - j, w = 6i - 2j
Polars and Vector Review Sheet
Pre-Calc. for AP Prep.
1)
θ
0
Name: ______________
Date: ______________
Graph: r = sin4 3θ + cos 2θ
r
θ
r
7
6
5
4
4
3
3
2
5
3
7
4
11
6

6

4

3

2
2
3
3
4
5
6
y
x








2

2)
θ
0

6

4

3

2
2
3
3
4
5
6

Graph: r = 3 sin2 θ cos θ
r
θ
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
r
y
x
Solve the following problems.
3)
A child throws a ball with a speed of 8 feet per second at an angle of 74° with the
horizontal. Express the vector described in terms of i and j. If exact values are not
possible, round components to 3 decimals.
4)
The magnitude and direction of two forces acting on an object are 35 pounds, N45°E, and
55 pounds, S30°E, respectively. Find the magnitude, to the nearest hundredth of a pound,
and the direction angle, to the nearest tenth of a degree, of the resultant force.
5)
Two forces, F1 and F2, of magnitude 60 and 70 pounds, respectively, act on an object.
The direction of F1 is N40°E and the direction of F2 is N40°W. Find the magnitude and
the direction angle of the resultant force. Express the direction angle to the nearest tenth
of a degree.
Find the magnitude v and direction angle θ, to the nearest tenth of a degree, for the given
vector v.
6) v = -3i - 4j
A) 5; 233.1°
B) 5; 53.1°
C) 5; 216.9°
D) 7; 233.1°
3) v = -5i + 12j
A) 13; 112.6°
B) 13; 67.4°
C) 15; 112.6°
D) 13; 157.4°