Download Pre-Calculus - Hood River County School District

Pre-Calculus
Chapter 8 – Vectors and Parametric Equations
Name
Period
8.1 and 8.2 Supplementary Worksheet
Use a ruler and a protractor (or make your best guess) to determine the magnitude (in centimeters)
and direction of each vector.
1.

p
2.

q


Use p and q above to find the magnitude and direction of each resultant. Use a ruler and a
protractor (or make your best guess). Sketch your method.
 

3. p + q
4. 2 p − 1 q
2

5. The vector v has a magnitude of 3.25 inches and a direction of 58°. Find the magnitude of its:
a) vertical component.
b)
horizontal component.
6. An airplane is flying due west at a velocity of 100 m/s. The wind is blowing out of the south at 5
m/s.
a) What is the magnitude of the airplane’s
resultant velocity?
b) What is the direction of the airplane’s
resultant velocity?
7. Find the ordered pair that represents the vector from A to B. Then find the magnitude of AB.
a) A( 3 , 1 ) to B( 0 , 8 )
b)
A( 2 , -1 ) to B( -3 , 4 )



8. Find an ordered pair to represent x in each equation if y   1, 4 and z  3,2 .
a)

 
x  4z  y
b)
  1
xz y
2
9. Find the magnitude of each vector. Then write each vector as a sum of unit vectors.
a)
8, - 6
b)
 7, - 5
10. Write AB as the sum of unit vectors for A and B.
a) A( 6 , 0 ) and B( 3 , -5 )
b)
A( 4 , -3 ) and B( 9 , -7 )
2



11. The vector u has a magnitude of 45 centimeters. If v   u , what is the magnitude of v ?
9
12. Jonah and Keone are working together to lift a box onto a ledge. Jonah is in the front on the ledge
and pulls with a force of 80 lbs at 50° with the horizontal. Keone is in the back on the floor and
pushes with a force of 100 lbs at 40° with the horizontal. What is the magnitude of the force
exerted on the box?
13. The Walker family is building a cabin for vacationing. Mr. Walker and his son Terrell have erected
a scaffold to stand on while they build the walls of the cabin. As they stand on the scaffold Terrell
pulls on a rope attached to a support beam with a force of 400 newtons (N) at an angle of 65° with
the horizontal. Mr. Walker pulls with a force of 600 N at an angle of 110° with the horizontal.
What is the magnitude of the combined force they exert on the log?
Pre-Calculus
Chapter 8 – Vectors and Parametric Equations
Name
Period
8.3 and 8.4 Supplementary Worksheet
1. Find the ordered triple that represents the vector from A to B. Then find the magnitude of AB.
a) A( 0 , -1 , -3 ) to B( -5 , 9 , 2 )
b)
A( -2 , 9 , 3 ) to B( -6 , -3 , 15 )



2. Find an ordered triple to represent x in each equation if y  1, - 3, - 8 and z  3, 9, - 1 .

 

 
a) x  3y  z
b) x  2y  5z
3.
Write AB as the sum of unit vectors for A and B.
a) A( -5 , -2 , 4 ) and B( 6 , -6 , 6 )
b)
A( -12 , 15 , -9 ) and B( -12 , 17 , -22 )
4. Find the inner product of a and b and state whether the vectors are perpendicular.




a) a  5 , 2 and b   3 , 7
b) a   4 , 7 , 8 and b  3 , 4 ,  2
5. Find the cross product and verify by using the dot product.


a) a  1 ,  3 , 2 and b   2 , 1 , 5


b) a  6 , 2 , 10 and b  4 , 1 , 9
6. A door is being pushed at a point 3 feet from the hinge. However, there is a heavy box blocking the
door so you apply 120 lbs of force to open the door. Assuming that you are pushing perpendicular to
the door, what is the torque being applied?
7. James is doing concentration curls against his knee using a 20 lb dumbbell. If the weight is 10 inches
from his elbow and he lifts it 70° to horizontal, then what is the torque about his elbow? Write the
answer in terms of foot-pounds.
8. Tikiro places a wrench on a nut and applies a downward force of 32 pounds to tighten the nut. If the
center of the nut is at the origin, the force applied at the point ( 0.65 , 0 , 0.3 ). Find the torque.
Pre-Calculus
Chapter 8 – Vectors and Parametric Equations
Name
Period
8.5 Supplementary Worksheet
1.

Find a vector that is perpendicular to both c  1 ,  3 , 2
indeed perpendicular.

and d  5 , 1 ,  2 and verify that it is
2. Find the magnitude and direction of the resultant vector for the diagram.
170 N
55°
300 N
3.
A 100N force and a 50N force act on the same object. The angle between the forces measures
90º. Find the magnitude of the resultant force and the angle between the resultant force and the
50N force.
4.
Denzel pulls a wagon along level ground with a force of 18N on the handle. If the handle makes
an angle of 40° with the horizontal, then find the horizontal and vertical components of the force.
5.
An 33-newton force acting at 90° and a 44-newton force acting at 60º act concurrently on an
object. What is the magnitude and direction of a third force that produces equilibrium on the
object?
6.
What would be the force required to push a 100 lb object along a ramp that is inclined 10° to the
horizontal?
7.
Alvaro works for a package delivery service. Suppose he is pushing a cart full of packages
weighing 100 lbs up a ramp 8 feet long at an incline of 25degrees. Find the work done by
gravity as the cart moves the length of the ramp. Assume that friction is not a factor.
Hint: Work done by gravity = Force x distance.
8.
A lighting system for a theater is supported equally by two cables suspended from the ceiling of
the theater. The cables form a 140° angle with each other. If the lighting system weighs 950 lbs,
what is the force exerted by each of the cables on the lighting system?