presentation mate vector

8/3/2019 Presentation Mate Vector

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NINA ALLYZA BT KEPOLSH. MAIZURA KASMAWATI SY. HAMZAH

NORHIDAYAH BT MOHD AMIN

SITI NOR AMIRA BT MOHMAD NOOR


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~To write vector.~To find magnitude of vectorand unit vector.~To do operation on vector.~To explain the concept ofvector.


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Vectors RepresentativeVectors may be represented by usingdirected linesegments orarrows.

The tail of the arrow is called the initial

point of

the vector and the head of the arrow is

theterminal point.

a, a

~

a

a

, ,


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The magnitude of a vector is represented by its

length and its direction is given by thedirection of the arrow.

MAGNITUDE OF

VECTOR


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The magnitude of =ai +bj

is v,22

bav

Example

Find the magnitude of v = 3i + 4j .

Magnitude of a vector

~

v


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Equal VectorsTwo vectors are equal if they have the samemagnitude and the same direction

Ifa is any nonzero vector, thena , the negative

ofa is defined to be the vector having the same

magnitude as a but oppositely directed.

Negatives Vectors

a

-a

a

a


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Two vector are equal if and only if they have thesame magnitude and direction. For example, in aparallelogram ABCD below,

AB=DCandAD=BCbutAB CD and BCDA

B

D

C

A


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Using the figure below, write each combinationof vectors as a single vector.

a)AC + CD b) DB + CB c)AC + CB

d)AD + DC + CB

AD

BC


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a)AD

b) DC

c)AB

d)AB


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A zero vector, denoted , is a vectorof length 0, and thus has all

components equal to zero. It is the

additive identity of the additivegroup of vectors. It has magnitudezero and does not have specific

direction.
http://mathworld.wolfram.com/Vector.htmlhttp://mathworld.wolfram.com/AdditiveIdentity.htmlhttp://mathworld.wolfram.com/AdditiveGroup.htmlhttp://mathworld.wolfram.com/AdditiveGroup.htmlhttp://mathworld.wolfram.com/AdditiveGroup.htmlhttp://mathworld.wolfram.com/AdditiveGroup.htmlhttp://mathworld.wolfram.com/AdditiveIdentity.htmlhttp://mathworld.wolfram.com/Vector.html


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If , and ,

prove that the points andare collinear and find the ratio

OA 6 a OB 3 a OC 4 a b

,A B C

:AB BC


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To determine the ratio of , thevectors and have to be found.

Using subtraction of vectors,

..(1)

:AB BCABBC

AB OB OA

3 6 b a

3( 2 )AB b a


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Similarly,

..(2)

BC OC OB

(4 ) 3 a b b

4 2 a b

2(2 )BC

a b


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From (1) and (2), and

respectively.

Thus, ,where is parallel to

and

12

3

AB b a

12

2BC a b

1

3 2AB CB

AB

BC1 1

3 2AB CB


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As the lines and are parallel and have a

common point , therefore and are collinear,

that is, they lie on the same straight line.

From , the ratio is obtained.

AB BC

B

,A B C

: 3: 2AB BC

1 13 2

AB BC


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When we multiply a vector by a scalar ,

then is a vector which direction depends upon the

sign of . If is positive, the direction of is thesame as that of vector .

Multiplication of a vector by a scalarv

v

vv

v

2v2v


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If a and b are any two vectors, then the sum a + b is thevector determined as follows :Position the vector b so that its initial point coincides with

the terminal point of a. The vector a + b is represented bythe arrow from the initial point of a to the terminal pointof b.

Addition Of Vectors

b

a

a+b

A

B

baAB

= resultant vector


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If u = , v = , express interm of and

a) u + vi j

2i j 2 i j


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a) u + v

2 2 2 2 i j i j i i j j=3 i j


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~Subtracting a vector is the same as

adding its negative.

~ The difference of the vectors p andq is the sum ofp and q.~p q = p + (q)


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Ifu = 4i+5jandv = i+3j

Find AB


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The scalar product(dot product) of two vectorsand is denoted by and defined as

a.b = |a||b| cos

Where is the angle between and

which converge to a point or diverge from a

point.

ba

a

b

a

b


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a b

Note:is an obtuse angle


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Let OA = a and OB =b

AB = OA + OB

AB = -a + b

AB = b - a


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Use this:a.a = a2

2 2 2

(AB) = ( ) + ( ) - 2( )( )cos

b-a . b-a = a.a + b.b 2 a b cos

b.b b.a b.a + a.a = a.a + b.b 2 a b cos2 b.a = 2 a b cos

a.b = a b cos

OA OB OA OB


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~ a.a = a2

~ a .b = b. A

~ a . (b + c) = a.b + a . C

~ m (a.b) = (ma) .b = (a.b)m

~ ( ) ( a b )c a b c) a b c


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~Evaluate

a) )43()2( ~~~~ kiji


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solution

a)

6

400132

432~~~~

kiji


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~Vectors are generally oriented on a coordinate system,the most popular of which is the two-dimensionalCartesian plane

~The Cartesian plane has a horizontal axis which islabeledx and a vertical axis labeledy


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OC, OA, OB are position vector that start fromorigin


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~ Some advanced applications of vectorsusing a three-

dimensional space, in which the axes arex,y, and z


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Example 1

Find vectorOA


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OA = OB + BA

OA = (3i) + (4j)

OA = 3i + 4j


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OAOBAB

AB ~~

32 ji ~~

4 ji

Find the vector AB


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Find vector for the following

question. Express in term of i

and j.

PQ


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OPOQPQ

PQ

~~54 ji

~~2ji


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cosbaba

Where is the angle betweenand which converge to a

point or diverge from a point.

a

b

a

b


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EXAMPLE

Find the angle between

and~~

3 jiv

~~64 jiu


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EXAMPLE

SOLUTION

2222

~~~~

6413

643

cos

cos

jiji

uvuv

uvuv


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13.142

130

9cos

130

9

520

185210

6143

1

thus


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Laws of multiplication of vector by a number.

I. 1 a = a , 0 a = 0 , m 0 = 0 , ( 1 ) a = a .II. m a = a m , | m a | = | m | | a |.III. m ( n a ) = ( m n ) a . ( Associative of

multiplication by a number).

IV. ( m + n ) a = m a + n a ,m ( a + b )= m a + m b .(Distributive of multiplication by a

number ).


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Thank you for lend your ears

See you again

Good luck for exam

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