# Chapter 7 - Applications of Vectors

Chapter 7.1 - Vectors as Forces

1.

Two forces of 20 N and 40 N act at an angle of 30 degrees to each other.  Determine the resultant of these two forces.

Answer:  approximately 58.2 N, in a direction 20.1 degrees rotated from the 20N force towards the 40 N force.

Video Solution to #1

2.

Kayla pulls on a rope attached to her sleigh with a force of 200N.  If the rope makes an angle of 20 degrees with the horizontal, determine

a) the magnitude of the force that pulls the sleigh forward;

b) the magnitude of the force that tends to lift the sleigh.

Answer:  a) approx. 187.9N;  b) approx. 68.4N

Video Solution to #2

3. 4.

A 75 kg mass is suspended from a ceiling by two lengths of rope that make angles of 50 degrees and 35 degrees with the ceiling.  Determine the magnitude of the tension in each rope.

Answer: approx. 474.7 N and 605.0 N

Video Solution to #4

5.

A sled is pointing north.  Jimmy pulls in a direction N 30 degrees E.  Ned pulls twice as hard in a direction N 40 degrees W.  In what direction will the sled move?

Answer: approx. N 18.6 degrees W.

Video Solution to #5

6.

Three forces of 7N, 9N and 12N are in equilibrium.  What is the angle between the two largest forces?

Video Solution to #6

Chapter 7.2 - Velocity

7.

An airplane is traveling at a velocity of 450 km/h [S 20 deg E] when it encounters a wind with a velocity of 140 km/h [N 40 deg E].  What is the ground velocity of the plane?

Answer:  approx. 398.9 km/h [S 37.7 deg E]

Video Solution to #7

8.

Ursula can swim at the rate of 9 m/s in still water.  She wants to swim to a point 2 km down the river.  In what direction should she head if the river is 10 km wide and the current moves at 3 m/s?

Answer: approx. 82.2 deg to the bank (but arguable 59.6 deg to the bank)

Video Solution to #8

9.

An airplane is traveling at a ground velocity of 320 km/h [N 15 deg W] in a wind-free environment.  The plane then encounters a wind which changes its ground velocity to 340 km/h [N 35 deg W].  What is the velocity of the wind?

Answer:  approx. 116.3 km/h [W 15.3 deg S]

Video Solution to #9

10.

An airplane is traveling 262 km/h [S 22 deg W] when it encounters a wind, changing its ground velocity to 270 km/h [S 35 deg W].  What is the velocity of the wind?

Answer:  approx. 60.7 km/h [W 21 deg N]

Video Solution to #10

11.

Phyllis swims downstream at a rate of 8m/s, at a 50 degree angle to the current which is traveling at 5 m/s.

a) How far downstream does she end up if the river is 500 m wide?

b) How long does it take her to cross the river?

Answer:  a) approx. 829 m      b)  approx. 81.6 seconds

Video Solution to #11

Section 7.3 - The Dot Product of Two Geometric Vectors

12. 13. 14.

15. Section 7.4 - The Dot Product of Algebraic Vectors

16 17. 18. 19. 20.

Triangle ABC has vertices A(4, -3, -1), B(-6, 1, 5) and C(2, 4, 4).  Determine the angles in the triangle

Answer:  approx. 48.5 degrees, 43.2 degrees, 88.3 degrees              Solution to #20

Chapter 7.5 - Scalar and Vector Projections

21. 22. 23. 24.

Determine a vector that makes an angle of 45 degrees with the x-axis, 90 degrees with the y-axis, and 135 degrees with the z-axis.

Answer:  (1, 0, -1) or any positive scalar multiple of (1, 0, -1)             Video Solution to #24

Chapter 7.6 - The cross product of two vectors

25. 26. 27.

Solve for k given that (6,2,3) X (1, -5, k) = (23, -21, -32)

Answer: k = 4                                             Video Solution to #27

28.

Solve for m given that the vectors (2, -4, 3) and (m, 2, -6) are both perpendicular to (-9, -21, -22).

Answer: m = 10                                         Video Solution to #28

Chapter 7.7 - Applications of the Dot Product and the Cross Product

29.

Calculate the amount of work done in each situation:

a)  A 100 kg man slides 10 m down a hill at an angle of 30 degrees to the horizontal

Answer:  approx. 4905 J                            Video Solution to #29a

b)  A snow shovel is pushed 3 m by a force of 500 N at an angle of 25 degrees to the horizontal

Answer:  approx. 1359 J                          Video Solution to #29b

30. 31. 32.

Parallelogram WXYZ has an area of 9 square units.  The vertices are W(-2,3,-9), X(2,5,a), Y (6,6,b) and Z (2,4,-6).  Solve for a and b.

Answer:  a = -4, b = -1  or   a = -130/17, b = -79/17                         Video Solution to #32

33.

A 50 N force is applied at the end of a wrench that is 75 cm long.  The force is applied at an angle of 60 degrees to the wrench.  Calculate the magnitude of the torque about the point of rotation.

Answer:  32.5 J                                                          Video Solution to #33