Find The Principal Unit Normal Vector, N ( T N(t N ( T ], For R ( T ) = T I + 6 T J R(t) = T \mathbf{i} + \frac{6}{t} \mathbf{j} R ( T ) = T I + T 6 ​ J .

Introduction

In the study of curves and surfaces, the principal unit normal vector plays a crucial role in understanding the direction of a curve at a given point. The principal unit normal vector, denoted by $N(t)$, is a vector that is perpendicular to the tangent vector of a curve at a point $t$. In this article, we will find the principal unit normal vector for the given curve $r(t) = t \mathbf{i} + \frac{6}{t} \mathbf{j}$.

The Curve $r(t)$

The curve $r(t)$ is defined by the parametric equations:

$$x(t) = t$$

$$y(t) = \frac{6}{t}$$

We can visualize this curve as a parametric curve in the $xy$-plane.

Finding the Tangent Vector

To find the principal unit normal vector, we first need to find the tangent vector of the curve. The tangent vector is given by the derivative of the position vector with respect to the parameter $t$:

$$\mathbf{r}'(t) = \frac{d}{dt} \left( t \mathbf{i} + \frac{6}{t} \mathbf{j} \right)$$

Using the product rule and the chain rule, we get:

$$\mathbf{r}'(t) = \mathbf{i} - \frac{6}{t^2} \mathbf{j}$$

Finding the Principal Unit Normal Vector

The principal unit normal vector is given by the formula:

$$N(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}$$

where $\|\mathbf{r}'(t)\|$ is the magnitude of the tangent vector.

First, we need to find the magnitude of the tangent vector:

$$\|\mathbf{r}'(t)\| = \sqrt{\left(1\right)^2 + \left(-\frac{6}{t^2}\right)^2}$$

Simplifying, we get:

$$\|\mathbf{r}'(t)\| = \sqrt{1 + \frac{36}{t^4}}$$

Now, we can find the principal unit normal vector:

$$N(t) = \frac{\mathbf{i} - \frac{6}{t^2} \mathbf{j}}{\sqrt{1 + \frac{36}{t^4}}}$$

Simplifying the Principal Unit Normal Vector

We can simplify the principal unit normal vector by rationalizing the denominator:

$$N(t) = \frac{\mathbf{i} - \frac{6}{t^2} \mathbf{j}}{\sqrt{1 + \frac{36}{t^4}}} \cdot \frac{\sqrt{1 + \frac{36}{t^4}}}{\sqrt{1 + \frac{36}{t^4}}}$$

Simplifying, we get:

$$N(t) = \frac{t^2 \mathbf{i} - 6 \mathbf{j}}{\sqrt{t^8 + 36}}$$

Conclusion

In this article, we found the principal unit normal vector for the given curve $r(t) = t \mathbf{i} + \frac{6}{t} \mathbf{j}$. We first found the tangent vector by taking the derivative of the position vector with respect to the parameter $t$. Then, we found the principal unit normal vector by dividing the tangent vector by its magnitude. Finally, we simplified the principal unit normal vector by rationalizing the denominator.

Applications of the Principal Unit Normal Vector

The principal unit normal vector has many applications in mathematics and physics. For example, it is used to find the curvature of a curve, which is a measure of how much the curve deviates from a straight line. It is also used to find the normal vector to a surface, which is a vector that is perpendicular to the surface at a given point.

Curvature of a Curve

The curvature of a curve is a measure of how much the curve deviates from a straight line. It is given by the formula:

$$\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}$$

where $\mathbf{r}'(t)$ is the tangent vector and $\mathbf{r}''(t)$ is the second derivative of the position vector.

Normal Vector to a Surface

The normal vector to a surface is a vector that is perpendicular to the surface at a given point. It is given by the formula:

$$\mathbf{n}(t) = \frac{\mathbf{r}'(t) \times \mathbf{r}''(t)}{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}$$

where $\mathbf{r}'(t)$ is the tangent vector and $\mathbf{r}''(t)$ is the second derivative of the position vector.

Conclusion

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Q: What is the principal unit normal vector?

A: The principal unit normal vector, denoted by $N(t)$, is a vector that is perpendicular to the tangent vector of a curve at a given point $t$. It is a unit vector that points in the direction of the normal to the curve at that point.

Q: How is the principal unit normal vector related to the tangent vector?

A: The principal unit normal vector is related to the tangent vector by the formula:

$$N(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}$$

where $\mathbf{r}'(t)$ is the tangent vector and $\|\mathbf{r}'(t)\|$ is the magnitude of the tangent vector.

Q: What is the significance of the principal unit normal vector?

A: The principal unit normal vector has several important applications in mathematics and physics. It is used to find the curvature of a curve, which is a measure of how much the curve deviates from a straight line. It is also used to find the normal vector to a surface, which is a vector that is perpendicular to the surface at a given point.

Q: How is the principal unit normal vector used in finding the curvature of a curve?

A: The principal unit normal vector is used in finding the curvature of a curve by the formula:

$$\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}$$

where $\mathbf{r}'(t)$ is the tangent vector and $\mathbf{r}''(t)$ is the second derivative of the position vector.

Q: How is the principal unit normal vector used in finding the normal vector to a surface?

A: The principal unit normal vector is used in finding the normal vector to a surface by the formula:

$$\mathbf{n}(t) = \frac{\mathbf{r}'(t) \times \mathbf{r}''(t)}{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}$$

where $\mathbf{r}'(t)$ is the tangent vector and $\mathbf{r}''(t)$ is the second derivative of the position vector.

Q: What are some common applications of the principal unit normal vector?

A: Some common applications of the principal unit normal vector include:

• Finding the curvature of a curve
• Finding the normal vector to a surface
• Calculating the arc length of a curve
• Finding the area of a surface

Q: How is the principal unit normal vector used in computer graphics?

A: The principal unit normal vector is used in computer graphics to create smooth and realistic curves and surfaces. It is used to calculate the normal vector to a surface, which is then used to determine the shading and lighting of the surface.

Q: How is the principal unit normal vector used in engineering?

A: The principal unit normal vector is used in engineering to calculate the stress and strain on a surface. It is used to determine the normal vector to a surface, which is then used to calculate the stress and strain on the surface.

Conclusion

In this article, we have discussed the principal unit normal vector and its applications in mathematics and physics. We have also answered some common questions about the principal unit normal vector and its uses in various fields.