Math 254 ~ Parametric Equations and Vector Review
Line through ( x1 , y1 ) and ( x2 , y2 ) :
x = x1 + t ( x2 − x1 ) , y = y1 + t ( y2 − y1 )
or
r (t ) =< x1 , y1 > +t < x2 − x1 , y2 − y1 >
Circle radius r centered at ( h, k ) :
x = h + r cos t , y = k + r sin t
or
r (t ) =< h, k > + r < cos t , sin t >
Definition – Length of a Vector
The length of the vector v =< v1 , v2 , v3 > is v = v12 + v22 + v32 .
Definition – Parallel Vectors
Two nonzero vectors u and v are parallel if there is some scalar c such that u = cv .
Definition – Dot Product
The dot product of u =< u1 , u2 , u3 > and v =< v1 , v2 , v3 > is
ui v = u1v1 + u2v2 + u3v3
Note: vi v = v
2
Definition – Orthogonal Vectors
The vectors u and v are orthogonal if ui v = 0 .
Theorem – Angle Between Two Vectors
If θ is the angle between two nonzero vectors u and v, then
cos θ =
ui v
u v
Definition – Cross Product of Two Vectors in Space
Let u = u1i + u2 j + u3k and v = v1i + v2 j + v3k be vectors in space. The cross product of
u and v is the vector
u × v = (u2 v3 − u3v2 )i − (u1v3 − u3v1 ) j + (u1v2 − u2 v1 )k
or
i
j
u × v = u1 u2
v1
v2
k
u3
v3
Theorem – Parametric Equations of a Line in Space
A line L parallel to the vector v =< a, b, c > and passing through the point ( x1 , y1 , z1 ) is
represented by the parametric equations
x = x1 + at , y = y1 + bt , and z = z1 + ct
Theorem – Standard Equation of a Plane in Space
The plane containing the point ( x1 , y1 , z1 ) and having a normal vector n =< a, b, c > can
be represented, in standard form, by the equation
a ( x − x1 ) + b( y − y1 ) + c( z − z1 ) = 0