ALTE DOCUMENTE
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Chapter 1 . Vectors and the Geometry of Space
Equations of Lines and Planes
Equations
of the line L through the point 
and parallel to
the
vector 
= 
where the vector is called the
direction
vector of the line L:
o
Vector
Equation: 
where 
= 
and 
= 
.
o
Parametric
Equations: 
, 
, 
.
o
Symmetric
Equations: 
= 
= 
.
o Two lines are skew lines if they do not intersect and are not parallel (and therefore do not lie on the same plane)
Equations
of the plane S through the point with normal
vector 
= that is orthogonal to
the plane S:
o
Vector
Equation: 
= 0 where = and = .
o
Scalar
Equation: 
+ 
+ 
= 0.
o
Linear
Equation: 
.
The
equation of a plane
through three points 
, 
and 
. Let's form vectors
and let the
normal vector 
.
Then the equation of a plane through three points P, Q and R is
Let
be the angle between two
planes with the normal vectors 
and
, respectively, then: 
.
o Two planes are parallel if their normal vectors are parallel.
o Two planes are perpendicular if their normal vectors are perpendicular.
The
distance D from a point 
to the plane 
can
be written in D = 
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