copyright © 2011 pearson, inc.. 6.1 day 1 vectors in the plane goal: apply the arithmetic of...
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Copyright © 2011 Pearson, Inc.
6.1 Day 1
Vectors in the Plane
Goal: Apply the arithmetic of vectors.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 3
What you’ll learn about
Two-Dimensional Vectors Vector Operations Direction Angles Applications of Vectors
… and whyThese topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.
Copyright © 2011 Pearson, Inc.
One vs. Two Quantities
Magnitude (Size) temperature distance Speed mass
Magnitude & Direction force velocity weight
Slide 6.1 - 4
Copyright © 2011 Pearson, Inc.
Vocabulary
Component Form:
Components:
Standard representation:
Zero vector:
Slide 6.1 - 6
Copyright © 2011 Pearson, Inc. Slide 6.1 - 8
Head Minus Tail (HMT) Rule
If an arrow has initial point x1, y
1 and terminal point
x2, y
2 , it represents the vector x2 x
1, y
2 y
1.
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Example 1: Showing Vectors are Equivalent
Show that the arrow from R = (-4, 2) to S = (-1, 6) is equivalent to the arrow from P = (2, -1) to Q = (5, 3).
Slide 6.1 - 9
Copyright © 2011 Pearson, Inc. Slide 6.1 - 10
Magnitude
If v is represented by the arrow from x1, y
1 to x2, y
2 ,then
v x2 x
1 2 y
2 y
1 2.
If v a,b , then v a2 b2 .
Copyright © 2011 Pearson, Inc. Slide 6.1 - 11
Example 2: Finding Magnitude of a Vector
Find the magnitude of v represented by PQ,
where P (3, 4) and Q (5,2).
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Example 3: Performing Vector Addition
Let u = and v = . Find u + v.
Slide 6.1 - 14
Copyright © 2011 Pearson, Inc.
Example 3: Performing Vector Operations
Let u = and v = . Find 2u.
2u - v
Slide 6.1 - 17
Copyright © 2011 Pearson, Inc. Slide 6.1 - 18
Exit Ticket Performing Vector Operations
Let u 2, 1 and v 5,3 . Find 3u v.
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6.1 Day 2
Vectors in the Plane
Goal: Use vectors to solve real-world problems.
Copyright © 2011 Pearson, Inc.
Example 5a: Finding the Components of a Vector
Find the components of the vector v with direction angle 115˚ and magnitude 6.
Slide 6.1 - 21
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Example 5b: Finding the Components of a Vector
Find the exact components of the vector v with direction
angle 30˚ and magnitude 8.
Slide 6.1 - 22
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Example 5c: Finding the Components of a Vector
Draw the indicated vector and show the components into which it is resolved.
A cannonball is launched with a speed of 170 m/s at 40° above the horizontal.
Slide 6.1 - 23
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Velocity and Speed
The velocity of a moving object is a vector
because velocity has both magnitude and
direction. The magnitude of velocity is ________.
________________ - the angle that a line of travel makes with due north, measured clockwise
Slide 6.1 - 24
Copyright © 2011 Pearson, Inc.
Example 7: Writing Velocity as a Vector
A DC-10 jet aircraft is flying on a bearing of 65˚ at 500 mph. Find the component form of the velocity of the airplane. Recall that the bearing is the angle that the line of travel makes with due north, measured clockwise.
Slide 6.1 - 25
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Calculate Magnitude and Direction.
Calculate the magnitude and direction of the vector.
Slide 6.1 - 26
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Example: Find the magnitude and direction angle of each vector.
a) b)
Slide 6.1 - 27
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Summary
Magnitude and direction
magnitude and direction
Slide 6.1 - 28
Copyright © 2011 Pearson, Inc.
Example: Calculating the Effect of Wind Velocity
A jet carrying Dora the Explorer is flying at 400 mph on a course with a bearing of 30º. If the jet experiences a crosswind blowing due south at 20 mph, find the resultant speed and direction of the jet. Round all values throughout the problem and the final answer to the nearest tenth.
Slide 6.1 - 32
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