**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.

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

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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

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.

6.

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

Answer: approx. 144.6 degrees

**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]

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)

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]

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]

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

**Section 7.3 - The Dot Product of Two Geometric Vectors**

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**Section 7.4 - The Dot Product of Algebraic Vectors**

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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**

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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**

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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

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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