# Vector calculus, normals to plane curves

• miglo
In summary, we are asked to show that n(t)=-g'(t)i+f'(t)j and -n(t)=g'(t)i-f'(t)j are both normal to the curve r(t)=f(t)i+g(t)j at the point (f(t),g(t)). After trying to find the unit normal of r(t), we can use the dot product to show that both n(t) and -n(t) are perpendicular to the tangent of the curve at (f(t),g(t)).
miglo

## Homework Statement

show that n(t)=-g'(t)i+f'(t)j and -n(t)=g'(t)i-f'(t)j are both normal to the curve r(t)=f(t)i+g(t)j at the point (f(t),g(t)).

## The Attempt at a Solution

i tried finding the unit normal of r(t) in hopes that it would be exactly what n(t) and its negative are, but after going through a lot of algebra i didnt get what i was hoping would be the solution, unless i did some algebra mistake. Anyways i doubt I am approaching this problem the right way and would just like a hint at what to do.

hi miglo!

hint: what is the tangent to f(t)i+g(t)j ?

f'(t)i+g'(t)j?

Yes. Are the supposed normals perpendicular to the tangent?

yeah i think i just figured it out like a minute ago
i take the dot product of r'(t) with n(t) which equals 0, therefore n(t) is normal to the curve at (f(t),g(t)) and then i do the same with -n(t)

## 1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with vector fields, which are quantities that have both magnitude and direction. It involves the study of differentiation and integration of vector fields, and their applications to physics and engineering.

## 2. What are normals to plane curves?

Normals to plane curves are lines that are perpendicular to the tangent line at a given point on a curve. They represent the direction in which the curve is changing most rapidly at that point.

## 3. How are vector calculus and normals to plane curves related?

Vector calculus is used to find the equations of normals to plane curves and to calculate their properties, such as length and curvature. It also helps in understanding the behavior of these curves and their applications in real-world problems.

## 4. What are some real-world applications of vector calculus and normals to plane curves?

Some common applications include calculating the forces acting on objects moving in a curved path, understanding the flow of fluids and heat, and analyzing the behavior of electric and magnetic fields. They are also used in computer graphics, robotics, and machine learning.

## 5. Is vector calculus difficult to learn?

As with any mathematical subject, vector calculus can be challenging to learn at first. However, with practice and a solid understanding of basic calculus and linear algebra, it can be mastered. There are also many online resources and textbooks available to help with learning vector calculus.

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