vectors lecture 1

7/27/2019 Vectors Lecture 1

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Vectors 1


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What is a vector?

A vector is a mathematical quantitywith two characteristics:

1) magnitude: how much, size of

vector, always a non-negativenumber, length of vector

2) Direction: orientation in space.

Can you think of science quantities

which are vectors in nature?

Students take notes:


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Vectors vs. scalar quantities

A scalar is a quantity which has only the

characteristic of magnitude.

Name some scalars in science...

Student answers/suggestions:


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In art!


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In navigation

New York Harbor


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Prop. #1: Geometrically, a vector

is represented as a rayV

The initial or tail end

The terminal orhead end


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Prop. #2: The length of the ray

represents magnitude and

direction of A is the angle the ray

makes with the +x axis

+X

y Vmagnitude

= 0o


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Prop. #3: Two vectors A and B

are equal if they have the same

magnitude and direction.

AB

This property allows us to move vectors around

on our paper/blackboard without changing

their properties.


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Prop. #4: A = -B says that vectors

A and B are anti-parallel.

They have same size but the

opposite direction.

A

B

A = -B also implies

B = -A


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Prop. 5: Vectors can be added

geometrically

Find A + B

A

B

Vector C is the sum ofA + B

C = A + B

O

A

BO

C

B


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Prop. #6: Vector Addition is

CommutativeA + B = B + A

VectorC is the sum ofA + B

C = A + B = B + A

This is the parallelogram method learned in trig.

Find A + B

A

BO

A

BO

C

B

A


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Prop. #7: Add vectors head to

headA

BC D

AB

C

D

S

S = A + B + C + D

O

could representfour forces

acting upon

point 0 -tug-of-war


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Prop. #8: Vector subtraction is

defined as:

A - B = A + (-B)

Find A - BA

B O-B

-B

D = A - B

A

B O -B

-B

D = A - BC = A + B

Note that the

magnitude ofD could be

larger than the magnitude

of the sum C.

B

A

B O


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Prop. #9: Let A be a vector and k

some scalar number. Then kA is

a vector with magnitude |k|A.The sign of k dictates the

direction of kA.

A2A

-3A


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A vectorA in the x-y plane can

be represented by its

perpendicular components

X axis

y axis A

AX

AY

Components AX and AY

can be positive, negative,

or zero. The quadrant

that vector A lies in

dictates the sign of thecomponents.

Components are scalars.


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When the magnitude of vector A

is given and its direction

specified then its componentscan be computed easily

X axis

y axis A

AX

AYAX = Acos

AY = Asin


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AX is negative while

AY is still positive

X axis

y axisA

AX

AY

AX = Acos

AY = Asin

The quadrant that Alies in dictates the

sign of the trig

functions.


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Magnitude and direction of a

vector can be found by knowing

its components

X axis

y axisA

AX

AY

tan = AY/AX = tan-1(AY/AX)

A (AX)2 (AY)2


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for this moment...

That is all for this slide file....


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vectors lecture 1

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