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Chapter 10: Problem 24

Find the principal unit normal vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+\mathbf{k}, \quad t=-\frac{\pi}{4} $$

Short Answer

Expert verified
The unit tangent vector and principal unit normal vector are key components in understanding a particle's motion along a path in space, which is characterized by the vector function \(r(t)\). This allows for a thorough understanding of path direction (described by the tangent vector) and how sharply a particle is turning at any given point (expressed by the principal unit normal vector). Hence the question involved finding these vectors.

Step by step solution

01

Differentiating Vector Function

The first step is to find the derivative of the given vector function \(r(t)\), which is defined as \(r'(t)\). Apply standard derivative rules component-wise to \(r(t) = \cos(t)\mathbf{i} + 2\sin(t)\mathbf{j} + \mathbf{k}\).
02

Evaluate r'(t) at the specified point

Now that we have the derivative, the next step is to evaluate \(r'(t)\) at \(t = -蟺/4\).
03

Find the magnitude of r'(t)

Next, it's time to find the magnitude of \(r'(t)\), denoted as \(||r'(t)||\). This is done by taking the square root of the sum of the squares of each component of the vector.
04

Compute Tangent Vector

Now, to find the tangent vector \(T(t)\), which is also called the unit tangent vector, we need to divide the derivative of the given vector function \(r'(t)\) by its magnitude.
05

Differentiate the Tangent Vector

Find the derivative of the tangent vector, denoted as \(T'(t)\) by applying the standard derivative rules component-wise shwon in step 4.
06

Compute Principal Unit Normal Vector

In the final step, to find the principal unit normal vector \(N(t)\), divide the derivative of the tangent vector \(T'(t)\) by its magnitude at the point \(t = -蟺/4\). This vector is referred to as the principal unit normal vector because it is orthogonal (or normal) to the tangent vector of the path of a point undergoing motion.

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Most popular questions from this chapter

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{2 t}{8+t^{3}} \mathbf{i}+\frac{2 t^{2}}{8+t^{3}} \mathbf{j} $$

Use the model for projectile motion, assuming there is no air resistance. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of \(45^{\circ}\) above the horizontal.

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True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { The velocity vector points in the direction of motion. } $$
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