geometric vectors. pre-calculus chapter 8 sections 1 & 2 2 of 22 vector a vector is a quantity...

Geometric Vectors

Pre-calculus Chapter 8 Sections 1 & 2

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Vector A vector is a quantity that has both direction and

magnitude.It is represented geometrical by a directed line segment.

So a directed line segment with initial point P and terminal point Q denoted by

Direction of the arrowhead show direction. The length represents the magnitude denoted by .a

.or PQa

a

P

Q


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• If a vector has it’s initial point at the origin, it is in standard position.

• The direction of the vector is the directed angle between the positive x-axis and the vector.

• The direction of is 45°.

• If both the initial point and the terminal point are at the origin, the vector is the zero vector and is denoted by .

• The magnitude is 0 and it can be any direction.

b

Vector

0


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• Two vectors are equal if and only if they have the same direction and the same magnitude.

• are equal since theyhave same direction and

• are equal.

• have the same direction but

• but they have different directions, so

Equal Vectors

yz

and .yz

uv

and

wx

and so ,wx

.wx

,yv

.yv


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• The sum of two or more vectors is call the resultant of the vectors. The follow will give the resultant:

Resultant Vectors (sum of vectors)


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• Find the sum

a. Copy then copy placing the initial points together.

Form a parallelogram and draw dashed lines for the other two sides.

The resultant is the vector from the vertex to the opposite vertex.

Example 1:using and wv

a. the parallelogram method

b. the triangle method

v

w


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• Find the sum

b. Copy then copy so the initial point of is on the terminal side of .

The resultant is the vector from the initial point of to the terminal point of .

Example 1 (cont) the parallelogram method

b. the triangle method:using and wv

v

w

w

v

v

w


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Opposite Vectors

Two vectors are opposites if they have the same magnitude and opposite directions.

are opposites, as are The opposite of You can use opposite vectors

to subtract vectors.

hg

and ji

and

. is gg

. is hghg


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Scalars A quantity with only magnitude is called a

scalar quantity. Mass, time, length and temperature

The numbers used to measure scalar quantities are called scalars.

The product of a scalar k and a vector is a vector with the same direction as and a magnitude of , if k > 0.

If k < 0, the vector has the opposite direction of and a magnitude of

a

a

ak

ak


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Example 2

Use the triangle method to find

Rewrite

Draw a vector twice the magnitude of

Draw a vector opposite direction and half the magnitude of

Put them tip to tail.

Connect the initial point from and terminal point of

.2

12 wv

.2

12

wv

.v

.w

v

.w


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Parallel Two or more vectors are said to be parallel if

and only if they have the same or opposite direction.

Two or more vectors whose sum is a given vector are called components of the given resultant vector. Often it is useful to have components that are

perpendicular.


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Example 3 A cruise ship leaves port and sails for 8o miles in a

direction of 50° north of due east. Draw a picture and find the magnitude of the vertical and horizontal components.

hyp

adj

hyp

opp cos and sin

miles 6150sin8080

50sin

y

y

miles 5150cos80

80

50cos

x

x

Algebraic Vectors


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Algebraic Vectors Vector can be represented algebraically using ordered

pairs of real numbers.The ordered pair (3, 5) can represent a vector in standard

position. Initial pt (0,0) and terminal pt (3,5).

Horizontal Mag. of 3, Vertical Mag. of 5.

Since vector of same mag. and direction are equal many vectors can be represented by the same ordered pair. In other words, a vector does not have to be in standard

position to use ordered pairs


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Ordered Pairs …Vectors Assume that P1 and P2 are any two points.

Drawing the horizontal and vertical components yields a right triangle.

So, the magnitude can be found by using the Pythagorean Theorem.


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Example 1Write the ordered pair that represents the vector from C(7, – 3) to D(–2, –1). Then find the magnitude of .CD

1212 , yyxxCD 212

212 yyxxCD

31 ,72 12 xx 12 yy

2 ,9

22 3172

22 29

85481 CD


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Vector Operations

When vectors are represented by ordered pair they can be easily added, subtracted or multiplied by a scalar.


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Example 2Let Find each of the following. .8,0 and 4,1 vu

vu a.

v

2

1 c.

vu b.

vu

32 d.

4,184,01

12,184,01

4,082

1,0

2

1

24 ,08,283 ,0242 ,12

16 ,2


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Example 3Radiology technicians Mary Jones and Joseph Rodriguez are moving a patient on an MRI machine cot. Ms. J is pushing the cot with a force of 120 newtons at 55° with the horizontal, while Mr. R. is pulling the cot with a force of 200 newtons at 40° with the horizontal. What is the magnitude of the force exerted on the cot?

12055cos 1x

SOH CAH TOAMr. R.

200 n.y2

x240°Ms. J.

120 n.

x1

y1

55°

12055sin 1y

20040cos 2x

20040sin 2y

98,69, 11 yx 129,153, 22 yx

227,22212998,15369 RJ

22 227222 RJ

n 5.317 RJ


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Unit Vector A vectors with a magnitude of one unit is called a unit

vector.

A unit vector in the direction of the x–axis is represented by

A unit vector in the direction of the y–axis is represented by So,

Any vectors can be expressed as

.i

.j

1,0 and 0,1 ji

21,aaa

.21 jaia


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Example 4Write as the sum of unit vectors for A(2, –7) and B(–1, 5).

First write as an ordered pair.

The write as the sum of unit vectors.

AB

AB

75 ,21 AB

12 ,3AB

AB

jiAB

123


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Daily Assignment Chapter 8 Sections 1 & 2 Text Book

Pg 491○#15 – 25 Odd;

Pg 497○#13 – 41 Odd;

geometric vectors. pre-calculus chapter 8 sections 1 & 2 2 of 22 vector a vector is a quantity...

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