Equation of a line in 3D
Finding the equation of a line when
you know
a I point and the direction gradient
g
A R line
a a a r A positionvector a is thepointonthe
line
0
vector b isthedirection vectorwhich
AB is parallelto b AR Xb isparallelto theCine
X is a scalar
R positionvector r is an
ATR r a arbitrarypointon theline
r a t AR
r a Xb r is thepositionrector
ofageneralpoint
Eg Find a vectorequation
ofthestraightlinewithpassthrough
Gi Sj t 4k and parallel to Ti 3k
r a t X b
r
, b 2 points
c
I R line Use TB as a direction vector
c d CD d C
9 ar
Youcan the one or D
o ofthe 2 points
andthe direction to form an equation
r for d t X d c
forthe line
Cartesian form of a line
a t kb
r can be expressed as
x a t Xb
Ff t x This forms b a tXb
3equations Z as 1X b3
Rearrange fork e
Finding the equation of a line when
you know
a I point and the direction gradient
g
A R line
a a a r A positionvector a is thepointonthe
line
0
vector b isthedirection vectorwhich
AB is parallelto b AR Xb isparallelto theCine
X is a scalar
R positionvector r is an
ATR r a arbitrarypointon theline
r a t AR
r a Xb r is thepositionrector
ofageneralpoint
Eg Find a vectorequation
ofthestraightlinewithpassthrough
Gi Sj t 4k and parallel to Ti 3k
r a t X b
r
, b 2 points
c
I R line Use TB as a direction vector
c d CD d C
9 ar
Youcan the one or D
o ofthe 2 points
andthe direction to form an equation
r for d t X d c
forthe line
Cartesian form of a line
a t kb
r can be expressed as
x a t Xb
Ff t x This forms b a tXb
3equations Z as 1X b3
Rearrange fork e
