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Equations of Lines and Planes in 3D Space: A Comprehensive Guide, Lecture notes of Engineering Mathematics

A comprehensive guide to understanding and deriving equations of lines and planes in three-dimensional space. It covers the fundamental concepts of direction vectors, normal vectors, parametric equations, and the relationship between points, lines, and planes. Illustrative examples and exercises to solidify understanding.

Typology: Lecture notes

2023/2024

Uploaded on 12/25/2024

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WEEK 3LECTURE I
Equationstflinesa
ndplanes
Lines Setup We will writethe equationof aline
teat passes through agivenpoint
Poko yo Zo and has the direction
of agiven vector VLabe
LI
2atag pkatagot
PitI PaffIIISEE Ie
0Q FTRC2a 2b 2e
fy
xPG ybelongstothe line
Xto rtr af
parametric ygot t.ae
equationof IZ
zott.cl
feline tparameter
Remart If we solve for above
yea Isgmffia.EE
pf3
pf4
pf5

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WEEK 3 LECTURE I

Equationstflinesandplanes

Lines Setup

We will^ write

the (^) equationof a^ line

teat passes through

a givenpoint

Poko

yo

Zo and^ has^ the^ direction

of a^ given

vector V^ L^ a^

b (^) e L I (^2) a

tag pkatagot

P

itI

Paff

III

SEE

Ie

0Q

FTRC2a

2b 2e

f

y

x PG

y belongs^

tothe^

line

X to r tr a

f

parametric y^ got

t.ae

equation

of IZ

zott.cl

feline

t parameter

Remart (^) If we^ solve^ for above

yea I

sgmffia.EE

Example

Write the (^) parametric equation^ of

the line

passing through the^ points^

PC 1,2 3 and

Q

2 y EE^ 1

Solution

yz.pe ziE7FaHeliue too first.ie 2 poit PCI zi if I direction^ vector 7 to^

v PQ 54 11,0^25237

QQ Cfo 2 i 6 2 57 a b^ C Equation of

the line^ ta

frm ez f z E

ta QC5p

2 t (^) parameter solution (^) point Q^ 5, 2

direction vector^

to Zo

u QP L^ I^ 5,2 0,3 1 2

us L^ 6,2 57 a b^ c

Ezof

line

X Etta

f (^) palamete E 2

PG

y z is^ in^ the^ plane naffest

be perpendicular

to top

hi

Tofte (^) zo n La^ b^ c TIP x^ X^

y yo^

Z Zo POTI (^) acx xoi (^) bcy foi CCZ ZT.iq n

eeg of^

the plane Hot b^ yo azo ax^ by t CZ

REMARI

let us think backwards

Given a plane with (^) equation Extle (^) y Ez^ D ex zx yi

zz

s (^) 7IE.gs E what is^ the^ relevance^ of

the Vector^ La^

b (^) c ex 2 I (^37) Answer (^) the vector^ v^ La^

le c will be^ a^ normal

vector to^ the plane with (^) equation ax by

cz d

Example

Find the^ equation^ of

the

plane containing the (^) points PG^ 2,^

Q O^2

2

and

R (^) C 3

O l

Solution

quoth

t

Q lo^ 2, I

point

PC 1 2 4

to (^) fo Zo normal vector^ to

plane

n (^) FR x^ poi PR L 3 l o^ C^2 I^47 4 2 3 PQ o 1 Z^ C 27 2 43 L^ l^ O^2

i j k

n FE^

5 g z I ti Ij t2 k n L^4 5 qe Ez (^) of (^) plane I Cx^ Ie

Cg

Ef E

C 2

1 If