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PROBLEM 1: What are (a) the x component and (b) the y component of a vector a in
the xy plane if its direction is θ ≔ 225 ⋅ deg counterclockwise from the positive direction of
the x axis and its magnitude is r ≔ 9.9 ⋅ m ? Make sure that you start with a picture.

rx ≔ r ⋅ cos ((θ)) = -7 m

ry ≔ r ⋅ sin ((θ)) = -7 m

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PROBLEM 2: The x component of a vector A is -18.2m and the y component is 43.4m. (a)
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What is the magnitude of A . (b) What is the angle between the resultant vector and the x-
axis.
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 ―→
PROBLEM 2: The x component of a vector A is -18.2m and the y component is 43.4m. (a)
―→
What is the magnitude of A . (b) What is the angle between the resultant vector and the x-
axis.

⎛ 43.4 ⎞
θrel ≔ atan ⎜―― ⎟ = 67.249 deg
⎝ 18.2 ⎠

θreal ≔ 180 ⋅ deg - θrel = 112.8 deg Note the subtracting

introduction the
additional sig fig

PROBLEM 3: A ship sets out to sail to a point 146 km due north. An unexpected storm
blows the ship to a point 127 km due east of its starting point. (a) How far and (b) in what
direction (as an angle from due east, where north of east is a positive angle) must it now
sail to reach its original destination?
Non-Commercial Use Only
PROBLEM 3: A ship sets out to sail to a point 146 km due north. An unexpected storm
blows the ship to a point 127 km due east of its starting point. (a) How far and (b) in what
direction (as an angle from due east, where north of east is a positive angle) must it now
sail to reach its original destination?

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PROBLEM 4: In the figure, a heavy piece of machinery is raised by sliding it a distance
d ≔ 12.2 ⋅ m along a plank oriented at angle θ ≔ 29.0 ⋅ deg to the horizontal. How far is it
moved (a) vertically and (b) horizontally?

dvert ≔ d ⋅ sin ((θ)) = 5.91 m

dhoz ≔ d ⋅ cos ((θ)) = 10.7 m

PROBLEM 5: A person walks in the following pattern: 3.0 km north, then 2.5 km west, and
finally 4.1 km south. (a) How far and (b) at what angle (measured counterclockwise from
east) would a bird fly in a straight line from the same starting point to the same final point?

Non-Commercial Use Only

 PROBLEM 5: A person walks in the following pattern: 3.0 km north, then 2.5 km west, and
finally 4.1 km south. (a) How far and (b) at what angle (measured counterclockwise from
east) would a bird fly in a straight line from the same starting point to the same final point?

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PROBLEM 6: Given two vectors a ＝ 4.0 ⋅ m ⋅ î - 3.0 ⋅ m ⋅ �� + 1.0 ⋅ m ⋅ k� and
―→
b ＝ -1.0 ⋅ m ⋅ î + 1.0 ⋅ m ⋅ �� + 4.0 ⋅ m ⋅ k� . (a) Draw a picture of these vectors. Computer in
―→―→ ―→―→ ―→―→―→
vector notation the following: (a) a + b , (b) a - b , (c) a - b + c ＝ 0
Non-Commercial Use Only
 ―→
PROBLEM 6: Given two vectors a ＝ 4.0 ⋅ m ⋅ î - 3.0 ⋅ m ⋅ �� + 1.0 ⋅ m ⋅ k� and
―→
b ＝ -1.0 ⋅ m ⋅ î + 1.0 ⋅ m ⋅ �� + 4.0 ⋅ m ⋅ k� . (a) Draw a picture of these vectors. Computer in
―→―→ ―→―→ ―→―→―→
vector notation the following: (a) a + b , (b) a - b , (c) a - b + c ＝ 0

―→ ―→
PROBLEM 7: In the figure, a vector a with a magnitude of || a || ＝ 22.2 ⋅ m is directed at
angle θ ≔ 77.0 ⋅ deg counterclockwise from the +x axis. What are the components (a) ax
and (b) ay of the vector? A second coordinate system is inclined by angle θrot ≔ 16.0 ⋅ deg
with respect to the first. What are the components
Non-Commercial Use(c) the new a'x and (d) a'y in this
Only
 ―→ ―→
PROBLEM 7: In the figure, a vector a with a magnitude of || a || ＝ 22.2 ⋅ m is directed at
angle θ ≔ 77.0 ⋅ deg counterclockwise from the +x axis. What are the components (a) ax
and (b) ay of the vector? A second coordinate system is inclined by angle θrot ≔ 16.0 ⋅ deg
with respect to the first. What are the components (c) the new a'x and (d) a'y in this
primed coordinate system?

θnew ≔ θ - θrot = 61 deg

ax' ≔ 22.2 ⋅ cos ⎛⎝θnew⎞⎠ = 10.8

ay' ≔ 22.2 ⋅ sin ⎛⎝θnew⎞⎠ = 19.4

a ≔ ‾‾‾‾‾‾‾‾
ax' 2 + ay' 2 = 22.2 check

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PROBLEM 8: Vector A has a magnitude of A ≔ 15.0 units (generic unknown unit) and
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another vector called B has a magnitude of B ≔ 7.30 units. The only other bit of
―→―→
information that you have is that their dot product A ⋅ B ＝ 14.0 . What is the angle
between the vectors? Non-Commercial Use Only
 ―→
PROBLEM 8: Vector A has a magnitude of A ≔ 15.0 units (generic unknown unit) and
―→
another vector called B has a magnitude of B ≔ 7.30 units. The only other bit of
―→―→
information that you have is that their dot product A ⋅ B ＝ 14.0 . What is the angle
between the vectors?

⎛ 14.0 ⎞
θresult ≔ acos ⎜―――― ⎟ = 82.7 deg
⎝ 15.0 ⋅ 7.30 ⎠

―→―→ ―→
PROBLEM 9: Consider the cross product F ＝ ⎛⎝ v ⨯ B ⎞⎠ . If you are given the following
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vectors below and you know that Bx ＝ By (magnitudes), find B ＝ Bx ⋅ î + By ⋅ �� + Bz ⋅ k�

Non-Commercial Use Only

 ―→―→ ―→
PROBLEM 9: Consider the cross product F ＝ ⎛⎝ v ⨯ B ⎞⎠ . If you are given the following
―→
vectors below and you know that Bx ＝ By (magnitudes), find B ＝ Bx ⋅ î + By ⋅ �� + Bz ⋅ ��

―→ ―→
F ＝ -112 ⋅ î + 14.0 ⋅ �� - 98.0 ⋅ �� v ＝ -6.0 ⋅ î + 8.0 ⋅ �� + 8.0 ⋅ ��

PROBLEM 10: You are given the following three vectors and from those you are asked to
―→―→ ―→
compute the following combinations of dot and cross products (a) d1 ⋅ ⎛⎝ d2 ⨯ d3 ⎞⎠ , (b)
―→―⎛→ ―→ ―→―→―→
d1 ⋅ ⎝ d3 ⨯ d2 ⎞⎠ , and (c) d1 ⋅ ⎛⎝ d2 + d3 ⎞⎠ .
Non-Commercial Use Only
 PROBLEM 10: You are given the following three vectors and from those you are asked to
―→―→ ―→
compute the following combinations of dot and cross products (a) d1 ⋅ ⎛⎝ d2 ⨯ d3 ⎞⎠ , (b)
―→―⎛→ ―→ ―→―→―→
d1 ⋅ ⎝ d3 ⨯ d2 ⎞⎠ , and (c) d1 ⋅ ⎛⎝ d2 + d3 ⎞⎠ .

―→ ―→
d1 ＝ -4.40 ⋅ î + 6.80 ⋅ �� + 5.80 ⋅ �� d2 ＝ -2.00 ⋅ î + -4.00 ⋅ �� + 2.00 ⋅ ��
―→
d3 ＝ 2.00 ⋅ î + 3.00 ⋅ �� + 1.00 ⋅ ��

Note that there are many ways to write a vector. For example we often write a vector as a
simple matrix. Below you can see you this is done for each of the 3 vectors. Where the first
entry is the x value , the second is the y value, and last entry is the z part.

⎡ -4.40 ⋅ m ⎤ ⎡ -2.00 ⋅ m ⎤ ⎡ 2.00 ⋅ m ⎤

d1 ≔ ⎢ 6.80 ⋅ m ⎥ d2 ≔ ⎢ -4.00 ⋅ m ⎥ d3 ≔ ⎢ 3.00 ⋅ m ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ 5.80 ⋅ m ⎦ ⎣ 2.00 ⋅ m ⎦ ⎣ 1.00 ⋅ m ⎦

⎡ -10 ⎤
((a)) d2 ⨯ d3 = ⎢ 6 ⎥ m 2
⎢ ⎥
⎣ 2⎦

d1 ⋅ ⎛⎝d2 ⨯ d3⎞⎠ = 96.4 m 3

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 ⎡ 10 ⎤
((b)) d3 ⨯ d2 = ⎢ -6 ⎥ m 2
⎢ ⎥
⎣ -2 ⎦

d1 ⋅ ⎛⎝d3 ⨯ d2⎞⎠ = -96.4 m 3

⎡ 0⎤
((c)) d2 + d3 = ⎢ -1 ⎥ m
⎢ ⎥
⎣ 3⎦

d1 ⋅ ⎛⎝d2 + d3⎞⎠ = 10.6 m 2

Non-Commercial Use Only