12 Vector Valued Functions

12.4 Unit Tangent and Normal Vectors

Unit Tangent Vector

Given a smooth vector-valued function r(t), we defined in Definition 12.2.4 that any vector parallel to r(t0) is tangent to the graph of r(t) at t=t0. It is often useful to consider just the direction of r(t) and not its magnitude. Therefore we are interested in the unit vector in the direction of r(t). This leads to a definition.

Definition 12.4.1 Unit Tangent Vector

Let r(t) be a smooth function on an open interval I. The unit tangent vector T(t) is

Example 12.4.1 Computing the unit tangent vector

Let r(t)=3cost,3sint,4t. Find T(t) and compute T(0) and T(1).

SolutionWe apply Definition 12.4.1 to find T(t).

T(t) =1r(t)r(t)

We can now easily compute T(0) and T(1): margin:
Figure 12.4.1: Plotting unit tangent vectors in Example 12.4.1. Λ


These are plotted in Figure 12.4.1 with their initial points at r(0) and r(1), respectively. (They look rather “short” since they are only length 1.)

In many ways, the previous example was “too nice.” It turned out that r(t) was always of length 5. In the next example the length of r(t) is variable, leaving us with a formula that is not as clean.

Example 12.4.2 Computing the unit tangent vector

Let r(t)=t2-t,t2+t. Find T(t) and compute T(0) and T(1).

SolutionWe find r(t)=2t-1,2t+1, and margin: -2246246xy Figure 12.4.2: Plotting unit tangent vectors in Example 12.4.2. Λ




When t=0, we have T(0)=-1/2,1/2; when t=1, we have T(1)=1/10,3/10. We leave it to the reader to verify each of these is a unit vector. They are plotted in Figure 12.4.2.

Unit Normal Vector

margin: xy Figure 12.4.3: Given a direction in the plane, there are always two directions orthogonal to it. Λ

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, we defined the normal line at a point to the be the line through the point that was perpendicular to the tangent line at that point. We can do a similar thing with vector-valued functions. Given r(t) in 2, we have 2 directions perpendicular to the tangent vector, as shown in Figure 12.4.3. It is good to wonder “Is one of these two directions preferable over the other?”

Given r(t) in 3, 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?”

The answer in both 2 and 3 is “Yes, there is one vector that is preferable.” Recall Theorem 12.2.5, which states that if r(t) has constant length, then r(t) is orthogonal to r(t) for all t. We know T(t), the unit tangent vector, has constant length. Therefore T(t) is orthogonal to T(t).

We’ll see that T(t) is more than just a convenient choice of vector that is orthogonal to r(t); rather, it is the “right” choice. Since all we care about is the direction, we define this newly found vector to be a unit vector.

margin: Note: T(t) is a unit vector, by definition. This does not imply that T(t) is also a unit vector. Λ
Definition 12.4.2 Unit Normal Vector

Let r(t) be a vector-valued function where the unit tangent vector, T(t), is smooth on an open interval I. The unit normal vector N(t) is

Example 12.4.3 Computing the unit normal vector

Let r(t)=3cost,3sint,4t as in Example 12.4.1. Sketch both T(π/2) and N(π/2) with initial points at r(π/2).

SolutionIn Example 12.4.1, we found T(t)=-35sint,35cost,4/5. Therefore

Figure 12.4.4: Plotting unit tangent and normal vectors in Example 12.4.3. Λ



We compute T(π/2)=-3/5,0,4/5 and N(π/2)=0,-1,0. These are sketched in Figure 12.4.4.

The previous example was once again “too nice.” In general, the expression for T(t) contains fractions of square-roots, hence the expression of T(t) is very messy. We demonstrate this in the next example.

Example 12.4.4 Computing the unit normal vector

Let r(t)=t2-t,t2+t as in Example 12.4.2. Find N(t) and sketch r(t) with the unit tangent and normal vectors at t=-1,0 and 1.

SolutionIn Example 12.4.2, we found


Finding T(t) requires two applications of the Quotient Rule:

T(t) =8t2+2(2)-(2t-1)(12(8t2+2)-1/2(16t))8t2+2,

This is not a unit vector; to find N(t), we need to divide T(t) by it’s magnitude.

T(t) =16(2t+1)2(8t2+2)3+16(1-2t)2(8t2+2)3


N(t) =14/(8t2+2)4(2t+1)(8t2+2)3/2,4(1-2t)(8t2+2)3/2

Because we are normalizing T(t), it is usually easier to scale it first. We see that T(t) is parallel to 2t+1,1-2t, which has length (2t+1)2+(1-2t)2=8t2+2, leading to the same N(t).

margin: -2246246xy Figure 12.4.5: Plotting unit tangent and normal vectors in Example 12.4.4. Λ

Using this formula for N(t), we compute the unit tangent and normal vectors for t=-1,0 and 1 and sketch them in Figure 12.4.5.

The final result for N(t) in Example 12.4.4 is suspiciously similar to T(t). There is a clear reason for this. If u=u1,u2 is a unit vector in 2, then the only unit vectors orthogonal to u are -u2,u1 and u2,-u1. Given T(t), we can quickly determine N(t) if we know which term to multiply by (-1).

Consider again Figure 12.4.5, where we have plotted some unit tangent and normal vectors. Note how N(t) always points “inside” the curve, or to the concave side of the curve. This is not a coincidence; this is true in general. Knowing the direction that r(t) “turns” allows us to quickly find N(t).

Theorem 12.4.1 Unit Normal Vectors in 2

Let r(t) be a vector-valued function in 2 where T(t) is smooth on an open interval I. Let t0 be in I and T(t0)=t1,t2 Then N(t0) is either


whichever is the vector that points to the concave side of the graph of r.

Application to Acceleration

Let r(t) be a position function. It is a fact (stated later in Theorem 12.4.2) that acceleration, a(t), lies in the plane defined by T and N. That is, there are scalars aT and aN such that


The scalar aT measures “how much” acceleration is in the direction of travel, that is, it measures the component of acceleration that affects the speed. The scalar aN measures “how much” acceleration is perpendicular to the direction of travel, that is, it measures the component of acceleration that affects the direction of travel.

We can find aT using the orthogonal projection of a(t) onto T(t) (review Definition 11.3.3 in Section 11.3 if needed). Recalling that since T(t) is a unit vector, T(t)T(t)=1, so we have


Thus the amount of a(t) in the direction of T(t) is aT=a(t)T(t). The same logic gives aN=a(t)N(t).

While this is a fine way of computing aT, there are simpler ways of finding aN (as finding N itself can be complicated). The following theorem gives alternate formulas for aT and aN.

margin: Note: Keep in mind that both aT and aN are functions of t; that is, the scalar changes depending on t. It is convention to drop the “(t)” notation from aT(t) and simply write aT. Λ
Theorem 12.4.2 Acceleration in the Plane Defined by T and N

Let r(t) be a position function with acceleration a(t) and unit tangent and normal vectors T(t) and N(t). Then a(t) lies in the plane defined by T(t) and N(t); that is, there exists scalars aT and aN such that



aT =a(t)T(t)=ddt(v(t))
aN =a(t)N(t)=a(t)2-aT2=a(t)×v(t)v(t)=v(t)T(t)

Note the second formula for aT: ddt(v(t)). This measures the rate of change of speed, which again is the amount of acceleration in the direction of travel.

  • Proof

    We see that

    a(t) =ddtv(t)=ddt(v(t)T(t))=(ddtv(t))T(t)+v(t)T(t)

    Since T(t) and N(t) are not parallel, this decomposition is unique and the coefficients tell us aT and aN.

    Because T=1, Theorem 12.2.5 tells us that T and T=TN are orthogonal. This means that


    Also, the Pythagorean theorem tells us that

Example 12.4.5 Computing aT and aN

Let r(t)=3cost,3sint,4t as in Examples 12.4.1 and 12.4.3. Find aT and aN.

SolutionThe previous examples give a(t)=-3cost,-3sint,0 and


We can find aT and aN directly with dot products:

aT =a(t)T(t)=95costsint-95costsint+0=0.
aN =a(t)N(t)=3cos2t+3sin2t+0=3.

Thus a(t)=0T(t)+3N(t)=3N(t), which is clearly the case.

What is the practical interpretation of these numbers? aT=0 means the object is moving at a constant speed, and hence all acceleration comes in the form of direction change.

Example 12.4.6 Computing aT and aN

Let r(t)=t2-t,t2+t as in Examples 12.4.2 and 12.4.4. Find aT and aN.

SolutionThe previous examples give a(t)=2,2 and


While we can compute aN using N(t), we instead demonstrate using another formula from Theorem 12.4.2.

aT =a(t)T(t)=4t-28t2+2+4t+28t2+2=8t8t2+2.
aN =a(t)2-aT2=8-(8t8t2+2)2=48t2+2.

When t=2, aT=1634 and aN=434. We interpret this to mean that at t=2, the particle is acculturating mostly by increasing speed, not by changing direction. As the path near t=2 is relatively straight, this should make intuitive sense. Figure 12.4.6 gives a graph of the path for reference. margin: -2246246t=2t=0r(t)xy Figure 12.4.6: Graphing r(t) in Example 12.4.6. Λ

Contrast this with t=0, where aT=0 and aN=4/2=22. Here the particle’s speed is not changing and all acceleration is in the form of direction change.

Example 12.4.7 Analyzing projectile motion

A ball is thrown from a height of 240ft with an initial speed of 64ft/s and an angle of elevation of 30. Find the position function r(t) of the ball and analyze aT and aN.

SolutionUsing Equation (12.3.1) of Section 12.3 we form the position function of the ball:


which we plot in Figure 12.4.7.

margin: 100200300100200t=0t=2t=3t=4xyt=1t=5 Figure 12.4.7: Plotting the position of a thrown ball, with 1s increments shown. Λ

From this we find

v(t)=64cos30,-32t+64sin30  and  a(t)=0,-32.

Computing T(t) is not difficult, and with some simplification we find


With a(t) as simple as it is, finding aT is also simple:


We skip finding N(t) and find aN through the formula aN=a(t)2-aT2 :


Figure 12.4.8 gives a table of values of aT and aN. When t=0, we see the ball’s speed is decreasing; when t=1 the speed of the ball is unchanged. This corresponds to the fact that at t=1 the ball reaches its highest point.

After t=1 we see that aN is decreasing in value. This is because as the ball falls, it’s path becomes straighter and most of the acceleration is in the form of speeding up the ball, and not in changing its direction. margin: t aT aN 0 -16 27.7 1 0 32 2 16 27.7 3 24.2 20.9 4 27.7 16 5 29.4 12.7 Figure 12.4.8: A table of values of aT and aN in Example 12.4.7. Λ

Our understanding of the unit tangent and normal vectors is aiding our understanding of motion. The work in Example 12.4.7 gave quantitative analysis of what we intuitively knew.

The next section provides two more important steps towards this analysis. We currently describe position only in terms of time. In everyday life, though, we often describe position in terms of distance (“The gas station is about 2 miles ahead, on the left.”). The arc length parameter allows us to reference position in terms of distance traveled.

We also intuitively know that some paths are straighter than others — and some are curvier than others, but we lack a measurement of “curviness.” The arc length parameter provides a way for us to compute curvature, a quantitative measurement of how curvy a curve is.

Exercises 12.4


Terms and Concepts

  1. 1.

    If T(t) is a unit tangent vector, what is T(t)?

  2. 2.

    If N(t) is a unit normal vector, what is N(t)r(t)?

  3. 3.

    The acceleration vector a(t) lies in the plane defined by what two vectors?

  4. 4.

    aT measures how much the acceleration is affecting the              of an object.


In Exercises 5–8., given r(t), find T(t) and evaluate it at the indicated value of t.

  1. 5.

    r(t)=2t2,t2-t, t=1

  2. 6.

    r(t)=t,cost, t=π/4

  3. 7.

    r(t)=cos3t,sin3t, t=π/4

  4. 8.

    r(t)=cost,sint, t=π

In Exercises 9–12., find the equation of the line tangent to the curve at the indicated t-value using the unit tangent vector. Note: these are the same problems as in Exercises 5.8..

  1. 9.

    r(t)=2t2,t2-t, t=1

  2. 10.

    r(t)=t,cost, t=π/4

  3. 11.

    r(t)=cos3t,sin3t, t=π/4

  4. 12.

    r(t)=cost,sint, t=π

In Exercises 13–16., find N(t) using Definition 12.4.2. Confirm the result using Theorem 12.4.1.

  1. 13.


  2. 14.


  3. 15.


  4. 16.


In Exercises 17–20., a position function r(t) is given along with its unit tangent vector T(t) evaluated at t=a, for some value of a.

  1. (a)

    Confirm that T(a) is as stated.

  2. (b)

    Using a graph of r(t) and Theorem 12.4.1, find N(a).

  1. 17.

    r(t)=3cost,5sint; T(π/4)=-334,534.

  2. 18.

    r(t)=t,1t2+1; T(1)=25,-15.

  3. 19.

    r(t)=(1+2sint)cost,sint; T(0)=25,15.

  4. 20.

    r(t)=cos3t,sin3t; T(π/4)=-12,12.

In Exercises 21–24., find T(t) and N(t).

  1. 21.


  2. 22.


  3. 23.

    r(t)=acost,asint,bt;  a>0

  4. 24.


In Exercises 25–30., find aT and aN given r(t). Sketch r(t) on the indicated interval, and comment on the relative sizes of aT and aN at the indicated t values.

  1. 25.

    r(t)=t,t2 on [-1,1]; consider t=0 and t=1.

  2. 26.

    r(t)=t,1/t on (0,4]; consider t=1 and t=2.

  3. 27.

    r(t)=2cost,2sint on [0,2π]; consider t=0 and t=π/2.

  4. 28.

    r(t)=cos(t2),sin(t2) on (0,2π]; consider t=π/2 and t=π.

  5. 29.

    r(t)=acost,asint,bt on [0,2π], where a,b>0; consider t=0 and t=π/2.

  6. 30.

    r(t)=5cost,4sint,3sint on [0,2π]; consider t=0 and t=π/2.

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