Copyright © 2011 Pearson, Inc.. 6.1 Day 1 Vectors in the Plane Goal: Apply the arithmetic of vectors

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.


One vs. Two Quantities

Magnitude (Size) temperature distance Speed mass

Magnitude & Direction force velocity weight

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Vector -

magnitude:

direction:

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Vocabulary

Component Form:

Components:

Standard representation:

Zero vector:

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Initial Point, Terminal Point, Equivalent


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.


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).

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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 .


Example 2: Finding Magnitude of a Vector

Find the magnitude of v represented by PQ,

where P (3, 4) and Q (5,2).


Vector Addition


Vector Addition


Example 3: Performing Vector Addition

Let u = and v = . Find u + v.

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Parallelogram Representation

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Scalar Multiplication

scalar:

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Example 3: Performing Vector Operations

Let u = and v = . Find 2u.

2u - v

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Exit Ticket Performing Vector Operations

Let u 2, 1 and v 5,3 . Find 3u v.


6.1 Day 2


Goal: Use vectors to solve real-world problems.


Resolving the Vector-


Example 5a: Finding the Components of a Vector

Find the components of the vector v with direction angle 115˚ and magnitude 6.

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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.

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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.

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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

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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.

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Calculate Magnitude and Direction.

Calculate the magnitude and direction of the vector.

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Example: Find the magnitude and direction angle of each vector.

a) b)

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Summary

Magnitude and direction

magnitude and direction

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6.1 Day 3


Goal: Calculate the resultant vector.


Resultant Vector

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Example: Add the Vectors to find the Resultant Vector

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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.

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: Combining Forces


Solution

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Documents

Copyright © 2011 Pearson, Inc.. 6.1 Day 1 Vectors in the Plane Goal: Apply the arithmetic of vectors