1. A vector function is a function that assigns to each x a vector
r
. Just like a regular function assigns a
point.
2. A vector function can be represented by a space curve. A space curve is just a plane curve in 3
dimensions. (A vector function can also draw out a plane curve.)
3. A smooth curve (in vector notation) is one whose corresponding vector function’s derivative with
respect to t never equals the zero vector. This is done to force the curve’s tangent vector to be
continuously turning. You find the tangent vector of a smooth curve at a point by taking the derivative of
the general vector equation with respect to t (by taking the derivative of each of its components, then
setting these as the components of a new vector), and finally plugging in the point into the component
equations.
Things to Know
:
When
r
t
=
c
(a constant),
r
is orthogonal to its derivative.
Distance formula in
R
3
:
x
h
2
y
k
2
z
l
2
.
Eqn. of a sphere:
x
h
2
y
k
2
z
l
2
=
r
2
Properties of vector addition/subtraction:
x
y
z
x
y
=
z
Standard representations:
i
=
1, 0, 0
j
=
0, 1, 0
k
=
0, 0, 1
Magnitude of Vector: place vector at origin, then use distance formula:
v
=
x
2
y
2
z
2
Dot product and associated formulas:
a
b
=
a
1
b
1
a
2
b
2
a
3
b
3
a
a
=
a
2
a
b
=
b
a
a
b
c
=
a
b
a
c
ca
b
=
c a
b
=
a
cb
0
a
= 0
a
b
=
a
b
cos
=
arccos
a
b
a
b
a
b
= 0.
b
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 Fall '07
 Bumpus
 Vector Space, Dot Product

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