Page 108 - Math Course 2 (Book 2)

Transformations: Using Vectors

Solve Problems Using Vectors

Example
CANOEING

CANOEING Suppose a person is canoeing due east across

Suppose a person is canoeing due east across a river at 4 miles per hour. If the current reduces
to half of its original speed, what is the resultant
a river at 4 miles per hour. If the river is flowing direction and velocity of the canoe?
south at 3 miles an hour, what is the resultant
direction and velocity of the canoe? Use scalar multiplication to find the magnitude of
the vector for the river.
The initial path of the canoe is due east, so a
vector representing the path lies on the positive
1
1
x-axis 4 units long. The river is flowing south, so a n| a | = | 3 | Magnitude of na; n = , | a | = 3
vector representing the river will be parallel to the 2 2
negative y-axis 3 units long. The resultant path can 3
be represented by a vector from the initial point of = or 1.5 Simplify.
2
the vector representing the canoe to the terminal
point of the vector representing the river. Next, use the Pythagorean Theorem to find the
magnitude of the resultant vector.
Use the Pythagorean Theorem.
Use the Pythagorean Theorem.
2
2
c = a + b 2 Pythagorean Theorem
2
2
c = a + b 2 Pythagorean Theorem
2
2
c = 4 + 3 2 a = 4, b = 3
2
2
c = 4 + 1.5 2 a = 4, b = 1.5
2
c = 25 Simplify.
2
c = 18.25 Simplify.
c = 25 Take the square root of each side.
c = 18.25 Take the square root of each side.
c = 5
c = 4.3
The resultant velocity of the canoe is 5 miles per
hour. Use the tangent ratio to find the direction of Then, use the tangent ratio to find the direction of
the canoe. the canoe.
3 1.5
tan � = Side opposite = 3, side adjacent = 4 tan � = Side opposite = 1.5, side adjacent = 4
4 4
3 1.5
� = tan –1 4 Solve for � � = tan –1 4 Solve for �
� ≈ 36.9 Use a calculator. � ≈ 20.6 Use a calculator.

The resultant direction of the canoe is about 36.9°
south of due east. If the current reduces to half its
original speed, the canoe travels
Therefore, the resultant Answer along a path approximately
Answer vector is 5 miles per hour at 20.6° south of due east at
about 4.3 miles per hour.
36.9° south of due east.

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