TRIG REVIEW PROBLEMS
Determine the missing angles and sides for the following triangles
1)
2)
3)
4)
5)
6)
7)
Find all the missing sides, angles, and diagonals for each triangle in the picture
8)
9)
10)
Problems 2-10 courtesy of www.MathWorksheetsgo.com
SYSTEMS OF EQUATIONS PROBLEMS
Solve for x and y in these systems on 2 equations
1)
2)
3)
4)
5)
6)
Solve for x,y, and z in these systems
7)
8)
9)
TRY SOLVING THESE PROBLEMS USING SUB. AND MATRIX METHODS
x + y + z = 25,000
0.06x + 0.07y + 0.08z = 1620
y – z = 16,000
The standard equation of a circle is x2 + y2 + Ax + By + C = 0. Find the equation of the circle
that passes through the points (3, -1), (-2, 4), and (6, 8).
Problems 1-9 courtesy of www.MathWorksheetsgo.com
Problem Set for Vector Review
Part 1
A)
Given a picture and the corresponding magnitude, convert these vectors into Cartesian vectors
⃑ = a𝑖̂ + b𝑗̂
of the form 𝑉
1)
⃑ | = 14 m/s
|𝑉
2)
|𝐹 | = 10 N
3)
|𝑎| = 9.81 m/𝑠 2
⃑ = a𝑖̂ + b𝑗̂ + c𝑘̂, then determine the length of
B)
Make a vector from A to B, express the vector as 𝑉
the vector.
1)
A= (2,3,4), B= (6,5,4)
2)
A= (1,0,0), B=(0,3,4)
3)
A= (-3,-4,6), B=(-1,-8,-8)
C)
Find the magnitude and direction associated with the following vectors. Express them in the
⃑ = |𝑉
⃑ |𝑢𝑣 ;
form 𝑉
1)
𝐹 = 2N𝑖̂ + 3N𝑗̂ – 6N𝑘̂
2)
⃑ = 4 m/s𝑖̂ – 6 m/s𝑗̂ + 7 m/s𝑘̂
𝑉
3)
𝐸⃑ = 12 N/C𝑖̂ + 5 N/C𝑗̂ – 13N/C𝑘̂
D)
Make a vector from point A to point B when given the 2 points. Find the magnitude of the vector
⃑ = |𝑉
⃑ |𝑢𝑣 ; 𝑉
⃑ = a𝑖̂ + b𝑗̂ + c𝑘̂
as well as the unit vector; express the vector in both of the following forms; 𝑉
1)
A= (1,2,3), B= (4,6,8)
2)
A= (0,1,0), B= (-2,3,-7)
3)
A=(-1,-2,9), B= (-1,4,-9)
Part 2
⃑
𝑈
⃑
𝑉
⃑⃑⃑
𝑊
A)
⃑,𝑉
⃑ , and 𝑊
⃑⃑⃑ , draw the following (don’t worry about perfect precision)
Given the vectors 𝑈
1)
⃑ +𝑉
⃑ -𝑊
⃑⃑⃑
𝑈
2)
⃑ -𝑉
⃑ +𝑊
⃑⃑⃑
2𝑈
3)
⃑ +𝑉
⃑ + 3𝑊
⃑⃑⃑
−2𝑈
B)
1)
2)
3)
⃑ =6𝑖̂ + 3𝑗̂ + 0𝑘̂, 𝑉
⃑ = -3𝑖̂ + 3𝑗̂ - 𝑘̂, and 𝑊
⃑⃑⃑ = -𝑖̂ + 5𝑗̂ + 2𝑘̂
Find 1, 2, and 3 from the previous part if 𝑈
Part C
(1) For A = (4i + 3j) and B = (-12i + 5j), draw the force polygon for A + B = C and find
C via analytical methods.
(2) a: If
, find the value of F to minimize the resultant force.
b: If F = 8 kN, find the value of  to minimize the resultant force.
(3) A Chevy Silverado and a VW Beatle must pull the seaplane onto dry land. If the Chevy provides 1500
lb of force, find the angle theta which will pull the seaplane directly backwards (along the vector F
shown) and minimizes the VW’s towing force. Find the resultant force vector.
Part D
(1) Vector A = (150i - 200j)lb; B = (200i + 480j) lb
(a) R = A + B
(b) R = 2A - B/2
(c) Find s so that R = sA + B has no j-component.
(2) The Space Shuttle’s ‘Canadarm’ is shown, along with a 2-d representation.
Determine the position vector from point A to point D.
⃑⃑⃑⃑𝑅 ) as well as all other forces acting on the
E)
Find the missing force when given the resultant (𝐹
object
⃑⃑⃑ if ⃑⃑⃑⃑
1) Find 𝑊
𝐹𝑅 = 0 (static equilibrium); if ⃑⃑⃑⃑
𝐹𝑅 = 3 kN downward.
y
⃑⃑⃑⃑
𝐹2 = 45 kN
4
3
x
⃑⃑⃑
𝐹1 = 27 kN
⃑⃑⃑ = ?
𝑊
2) Find ⃑⃑⃑
𝐹3 if ⃑⃑⃑⃑
𝐹𝑅 = 6 lbs𝒊̂ + 3 lbs𝒋̂; ⃑⃑⃑
𝐹1 = 13 lbs𝒊̂ - 4 lbs𝒋̂; ⃑⃑⃑
𝐹2 has a magnitude of 20 lbs and makes a
35⁰ angle with the positive x-axis going counterclockwise.
⃑⃑⃑1 = -17 lbs𝒊̂ - 16 lbs𝒋̂; 𝐹
⃑⃑⃑2 has a magnitude of 75 lbs and makes
3) Find ⃑⃑⃑
𝐹3 if ⃑⃑⃑⃑
𝐹𝑅 = 56 N𝒊̂ + 30 N𝒋̂; 𝐹
a 65⁰ angle with the positive y-axis going counterclockwise.
PART 3
⃑ , θ, and the components of 𝐴 parallel to and perpendicular to
A) Find the following: 𝐴 • 𝐵
⃑.
𝐵
⃑ = 2𝑖̂ + 9𝑗̂ + 2𝑘̂
1) 𝐴= 3𝑖̂ + 5𝑗̂ + 𝑘̂; 𝐵
⃑ = 3𝑖̂ - 7𝑗̂ - 𝑘̂
2) 𝐴= -4𝑖̂ - 5𝑗̂ + 2𝑘̂; 𝐵
⃑ = 14𝑖̂ - 10𝑗̂ + 2𝑘̂
3) 𝐴= 7𝑖̂ - 5𝑗̂ + 𝑘̂; 𝐵
̂ - 4𝑘̂; 𝐵
⃑ = 2𝑖̂ + 𝑗̂ + 2𝑘̂
4) 𝐴= 3𝑖̂ + 2𝑗
B)
Dot Product
(1) A street vendor sells a hamburgers, b hot dogs, and c soft drinks; he charges $2 per
hamburger, $1.50 per hot dog, and $1 per soft drink. If A = (ai + bj + ck) and P = (2i
ur ur
+ 1.5j + 1k), what is the meaning of A  P ?
(2) For what values of a are the vectors (-6i + aj + 2k) and (ai + a2j + ak) orthogonal?
(3) Find the work done by a force F = (8i - 6j + 9k)N as it moves an object from point
(0,10,8)m to (6,12,20)m.
(4) A sailboat travels south with a wind direction of south-40 degrees east. If the wind
provides a 400 lb force on the sail, find the work done on the boat over a distance of
120 feet.
(5) If the force component perpendicular to the gearshift lever (with position vector r)
must be 5 N to shift gears, find the magnitude of the force applied in the direction of unit
vector F.
Part 4
⃑ , the angle between the 2 vectors, as well as 𝐵
⃑ × 𝐴
A) Find 𝐴 × 𝐵
⃑ = 2𝑖̂ + 9𝑗̂ + 2𝑘̂
1) 𝐴= 3𝑖̂ + 5𝑗̂ + 𝑘̂; 𝐵
⃑ = 3𝑖̂ - 7𝑗̂ - 𝑘̂
2) 𝐴= -4𝑖̂ - 5𝑗̂ + 2𝑘̂; 𝐵
⃑ = 14𝑖̂ - 10𝑗̂ + 2𝑘̂
3) 𝐴= 7𝑖̂ - 5𝑗̂ + 𝑘̂; 𝐵
̂ - 4𝑘̂; 𝐵
⃑ = 2𝑖̂ + 𝑗̂ + 2𝑘̂
4) 𝐴= 3𝑖̂ + 2𝑗
B)
Determine where the magnetic force vector will be oriented (to the left,right;
⃑
up,down; into or out of the page) 𝐹𝐵 = q𝑣 × 𝐵
1) If q = 1 and
⃑
𝐵
𝑣
2) If q = -1 and
⃑ 𝑖𝑛
𝐵
𝑣
3) If q = 1 and
𝑣
⃑
𝐵