Vectors

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Lesson 9-6

• Vectors

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Transparency 9-6
5-Minute Check on Lesson 9-5

• Determine whether the dilation is an enlargement,
  a reduction or a congruence transformation based
  on the given scaling factor.
• r ? 2. r - 4
  3. r 1
• Find the measure of the dilation image of AB with
  the given scale factor
• 4. AB 3, r - 2 5.
  AB 3/5, r 5/7
• 6.
  Determine the scale factor of the dilated image

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Standardized Test Practice
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B
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Click the mouse button or press the Space Bar to
display the answers.
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Objectives

• Find magnitudes and directions of vectors
• Perform translations with vectors

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Vocabulary

• Vector a quantity that has both magnitude and
  direction
• Vector magnitude its length (use distance
  formula to find magnitude)
• Vector direction measure of angle the vector
  forms with x-axis
• Component form ordered pair representation of a
  vector lt?x, ?ygt
• Resultant the sum of two vectors
• Scalar a positive constant
• Scalar multiplication multiplying a vector by a
  scalar

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Vectors
Magnitude is found by using the distance
formula Vector addition add corresponding
components Scalar multiplication multiply each
component by the scalar value. Negative value
reverses direction
Direction is measured in reference to the x-axis.
It is usually found using tan-1 . positive
x-axis is 0 positive y-axis is 90 negative
x-axis is 180 negative y-axis is 270 In
navigation, we use north as 0 degrees and go
around clock-wise.
? Magnitude ?
x - Direction
Most practical aspect for physics and engineering
classes wind velocity, friction, drag, lift,
etc can all be represented by vectors
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Example 6-1a
Find the change of x values and the corresponding
change in y values.
Component form of vector
Simplify.
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Example 6-1b
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Example 6-2a
Find the magnitude.
Distance Formula
Simplify.
Use a calculator.
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Example 6-2a
Sub
Simplify.
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Example 6-2a
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Example 6-2b
Answer ? 5.7 225
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Example 6-3a
First graph quadrilateral HJLK.
Answer
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Example 6-3b
Answer
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Example 6-5a
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the river
is flowing south at 3 miles an hour, what is the
resultant direction and velocity of the canoe?
The initial path of the canoe is due east, so a
vector representing the path lies on the positive
x-axis 4 units long. The river is flowing south,
so a vector representing the river will be
parallel to the negative y-axis 3 units long. The
resultant path can be represented by a vector
from the initial point of the vector representing
the canoe to the terminal point of the vector
representing the river.
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Example 6-5a
Use the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Take the square root of each side.
The resultant velocity of the canoe is 5 miles
per hour.
Use the tangent ratio to find the direction of
the canoe.
Use a calculator.
Answer Therefore, the resultant vector is 5
miles per hour at 36.9 south of due east.
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Example 6-5a
CANOEING 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 direction and velocity of
the canoe?
Use scalar multiplication to find the magnitude
of the vector for the river.
Simplify.
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Example 6-5a
Next, use the Pythagorean Theorem to find the
magnitude of the resultant vector.
Pythagorean Theorem
Simplify.
Take the square root of each side.
Then, use the tangent ratio to find the direction
of the canoe.
Use a calculator.
Answer If the current reduces to half its
original speed, the canoe travels along a path
approximately 20.6 south of due east at about
4.3 miles per hour.
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Example 6-5b

• KAYAKING Suppose a person is kayaking due east
  across a lake at 7 miles per hour.
• a. If the lake is flowing south at 4 miles an
  hour, what is the resultant direction and
  velocity of the canoe?
• b. If the current doubles its original speed,
  what is the resultant direction and velocity of
  the kayak?

Answer Resultant direction is about 29.7 south
of due east resultant velocity is about 8.1
miles per hour.
Answer Resultant direction is about 48.8 south
of due east resultant velocity is about 10.6
miles per hour.
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Summary Homework

• Summary
• A vector is a quantity with both magnitude and
  direction
• Magnitude is the distance between the two
  component vectors (Pythagorean Thrm)
• Vectors can be used to translate figures on the
  coordinate plane
• Homework
• pg 503-504 15-17, 24-28, 47-50