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Mathematical Expression Editor

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We introduce two important unit vectors.

Given a smooth vector-valued function , any vector parallel to is tangent to the
graph of at . It is often useful to consider just the direction of and not its
magnitude. Therefore we are interested in the unit vector in the direction of . This
leads to a definition.

Let be a smooth function on an open interval . The unit
tangent vector is

Let . Find .

The unit tangent vector always has a constant magnitude of .

In previous courses, we found tangent lines to curves at given points. Just as knowing
the direction tangent to a path is important, knowing a direction orthogonal to a
path is important. When dealing with real-valued functions, one defines the normal
line at a point to the be the line through the point perpendicular to the
tangent line at that point. We can do a similar thing with vector-valued
functions. Given in , we have directions perpendicular to the tangent vector

The young mathematician wonders “Is one of these two directions preferable over the
other?” This question only gets harder in higher dimensions. Given in , there are
infinite vectors orthogonal to the tangent vector at a given point. Again, we might
wonder “Is one of these infinite choices preferable over the others? Is one of these the
‘right’ choice?” Well, we have several options for finding vectors normal to curves. In
if we could write the tangent vector as: and then a normal vector as for a vector
normal to . You can check for yourself that this vector is normal to using the dot
product. In two-dimensions, the vector defined above will always point
“outward” for a closed curve drawn in a counterclockwise fashion. Below we see a
closed curve drawn in a counterclockwise fashion with some normal vectors:

On the other hand, there is no analogous trick for vector-valued functions in higher
dimensions. In this case, recall:

If has constant length, then is orthogonal to for all .

Since is a unit tangent vector, it necessarily has constant length. Therefore

Even though is a unit vector, this does not imply that is also a unit
vector.

Since constructing normal vectors using the derivative works in all dimensions, we
will often use this to construct our unit normal vector.

Let be a vector-valued
function where the unit tangent vector, , is smooth on an open interval . The
unit normal vector is Some folks call this the principal unit normal
vector.

The principal unit normal vector will always point toward the “inside” of how a curve
is curving.

Let as before. Find the principal unit normal vector .

As a gesture of friendship,
we present you with the following graph of the situation.