NHAA/IMK/UNIMAP

VECTORS

DOT PRODUCT

CROSS PRODUCT

APPLICATIONS

NHAA/IMK/UNIMAP

DIRECTIONS OF ANGLES & DIRECTIONS OF COSINES

,, - Are the angles that the vector OP makes with positive axis- Knows as the direction angles of vector OP

DIRECTION OF COSINES

180,,0;cos

cos

cos

OP

z

OP

y

OP

x

NHAA/IMK/UNIMAP

Example 1Find the direction cosines and direction angles of:

3,2,11,1,0

3,1,2

QPii

ui

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DOT PRODUCT Also known as inner product or scalar product The result is a scalar

If and then:321 ,, uuuu 321 ,, vvvv

332211

321321

,,.,,.

vuvuvu

vvvuuuvu

NHAA/IMK/UNIMAP

Example 2If and :

2 9

i u v

ii u v u

1,3,2 u 3,2,0 v

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DOT PRODUCT Angle Between 2 Vectors

u

v vu

vu.cos

If the vectors lies on the same line or parallel to each other, then

0u v u

v

NHAA/IMK/UNIMAP

Example 3Find the angles between and

6,0,43,0,2

4,3,23,2,1

vuii

vui

u v

NHAA/IMK/UNIMAP

DOT PRODUCTProperties of Dot Product

NHAA/IMK/UNIMAP

CROSS PRODUCT The result is a vector

If and then: (determinant of the matrix)

321 ,, uuuu 321 ,, vvvv

kvuvujvuvuivuvu

vvv

uuu

kji

vu

122113312332

321

321

NHAA/IMK/UNIMAP

Example 3Find the cross product between and

6,0,43,0,2

4,3,23,2,1

vuii

vui

u v

NHAA/IMK/UNIMAP

CROSS PRODUCTProperties of Cross Product jkiijkkijix

jikikjkjiviii

kkjjiivii

vuvuvi

vuv

auvi

vkuvukvukiii

wuvuwvuii

uvvui

,,

,,

0

sin

0

000

if u and v are parallel

NHAA/IMK/UNIMAP

APPLICATIONS1. PROJECTIONSThe vector projection of u = onto a nonzero vector Is the vector determined by dropping a perpendicular from Q to the line PS.

PQ PSv

PR

u

v

Q

P R S

uprojv

NHAA/IMK/UNIMAP

APPLICATIONS1. PROJECTIONS• Scalar projection• The magnitude of the vector projection:

• Vector projection

v

vuucompv

.

v

vucompuproj vv

NHAA/IMK/UNIMAP

APPLICATIONS1. AREA OF TRIANGLE & PARALLELOGRAMThe magnitude of uxv is the area of parallelogram

u

vsinvh

vu

vu

heightbase

sin

ramparallelog of area

NHAA/IMK/UNIMAP

APPLICATIONS1. AREA OF TRIANGLE Half of the area of parallelogram

vu

vu

2

1

sin2

1

ramparallelog of area2

1 triangleof area

v

u

NHAA/IMK/UNIMAP

APPLICATIONS

i. Parametric Equations of a Line in

ii. Equations of Planes

iii. Distance from a Point to the Plane

3R

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APPLICATIONSiii. Lines & Line Segment in SpaceParametric Equations

z

y

x

P0(x0,y0,z0) LP(x,y,z)

v

Line L is the set of all points P(x,y,z) for which parallel to :

PP0 v

RtvtPP ,0

NHAA/IMK/UNIMAP

APPLICATIONSTherefore the parametric equation for L :

ctzzctzz

btyybtyy

atxxatxx

kcjbiatkzzjyyixx

vtPP

00

00

00

000

0

ParametricEquation

NHAA/IMK/UNIMAP

Cartesian equation:

c

zz

b

yy

a

xxt 000

NHAA/IMK/UNIMAP

Example 3:Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to 2,4,2 v

NHAA/IMK/UNIMAP

Example 4:Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4)

NHAA/IMK/UNIMAP

APPLICATIONSDistance from point S to line L

S

P v

sinPS

From the properties of Cross Product

v

vuu

vuvu

sin

sin

Formula of Distance from point S to L

v

vPSD

L

u

NHAA/IMK/UNIMAP

Example 5:Find the distance from the point S(1,1,5) to the line

tztytxL 2,3,1:

NHAA/IMK/UNIMAP

APPLICATIONSLines of Intersection

n1

n2

v

Line of intersection

21 nnv Finding v :

NHAA/IMK/UNIMAP

Example 8:Find a vector parallel to the line of intersection of the planes

522

15263

zyx

zyx

NHAA/IMK/UNIMAP

Example 9:Find the parametric equation for the line in which the planes

Intersect.

522

15263

zyx

zyx

NHAA/IMK/UNIMAP

Example 9:Find the point where the line

Intersects the plane

tztytx 1,2,23

8

6623 zyx

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APPLICATIONSEquation of Planes

P0(x0,y0,z0)P(x,y,z)

n

Vector is on the plane M and vector which is perpendicular to M known as normal vector, n

PP0

From the properties of Dot Product

0.0 nPP

NHAA/IMK/UNIMAP

APPLICATIONSEquation of Planes vun

Normal vector n :

u

v

vun

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APPLICATIONSLet and 0000 ,,,,, zyxPcban zyxP ,,

dczbyax

czbyaxczbyax

cbazzyyxx

nPP

000

000

0

0,,.,,

0.

EQUATION OF PLANE

NHAA/IMK/UNIMAP

Example 6:Find an equation of plane through P0(-3,0,7) perpendicular to 1,2,5 n

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Example 7:Find equation of plane through 3 points:

0,3,00,0,21,0,0 CBA

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APPLICATIONSDistance from a Point to the Plane

P0

P

n D

n

nPPD

.0

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APPLICATIONS

222

000

0

,,

,,,,

.

cba

dczbyax

cba

cbazzyyxx

n

nPPD

Equation of plane,With n = <a,b,c>

NHAA/IMK/UNIMAP

Example 10:Find the distance from S(1,1,3) to the plane 6623 zyx