# Calculus 10.7 (Vector Functions &amp; Space Curves)

A vector function is simply a function whose domain is a set of real numbers and whose range is a set of vectors
Vector function (definition)
The component functions of this vector equation are,
f(t) = t³
g(t) = ln(3-t)
h(t) = √t
and the domain of r is the interval,
[0,3)
since ln(0) is undefined and the square root of a negative number is also undefined
What are the component functions of the vector function,
r(t) =
what is the domain of r?
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Calculus 10.7 (Vector Functions &amp; Space Curves)
or any similar topic only for you The limit of a vector function is defined by taking the limits of its component functions
How is the limit of a vector function defined? The limit of r is the vector whose components are the limits of the component functions of r.
Find the lim r(t) as t approaches 0 where,
r(t) = (1+t³)i + (te⁻¹)j + (sint/t)k
The corresponding parametric equations are,
x = 1 + t
y = 2 + 5t
z = ⁻1 + 6t
Note that this means that these are parametric equations of a line passing through the point
(1,2,⁻1) and parallel to the vector
<1,5,6> according to section 10.5
Describe the curve defined by the vector function,
r(t) = <1+t,2+5t,⁻1+6t> Sketch the curve whose vector equation is,
r(t) = (cost)i + (sint)j + (t)k Use the line segment equation from 10.5
Find a vector equation and parametric equations for the line segment that joins the point,
P(1,3,⁻2)
to the point,
Q(2,⁻1,3)
Find a vector function that represents the curve of intersection of the cylinder,
x² + y² = 1
and the plane,
y + z = 2

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