# Proving Vector A and d\vec{}A/dt Are Perpendicular With Constant Magnitude

In summary, The problem asks for the proof that if a vector \vec{}A has constant magnitude, then it is perpendicular to its time derivative d\vec{}A/dt, given that d\vec{}A/dt is not equal to 0. This can be illustrated by imagining a circular motion with a constant radius, where the two vectors with magnitude A are displaced by an amount delta theta. Taking the limit of the vector difference delta A divided by delta theta as delta theta approaches 0 proves the perpendicularity. An example of this problem can be seen in a circular motion with constant radius.

## Homework Statement

If $$\vec{}A$$ has constant magnitude show that $$\vec{}A$$ and d$$\vec{}A$$/dt are perpendicular provided that d$$\vec{}A$$/dt $$\neq$$0

also give one physical example of this problem.

thnx.

## The Attempt at a Solution

This can be thought of as a circular motion problem with constant radius. Since the radius vector has a constant magnitude (constant radius), then only the position of the vector changes with time. Draw two vectors with magnitue A, one displaced an amount delta theta from the other. Find the vector difference which will be delta A, then take the limit of delta A divided by delta theta as delta theta goes to zero.

i didnt got u !

i didnt got u !

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ooops sorry !

I didnt knew the rules..
it won't happen next tym.

The magnitude of A squared is A.A (dot product). Is that a help?

## 1. How do you prove that Vector A and d\vec{}A/dt are perpendicular?

To prove that Vector A and d\vec{}A/dt are perpendicular, we can use the dot product method. If the dot product of two vectors is equal to zero, then the vectors are perpendicular. So, we can take the dot product of Vector A and d\vec{}A/dt and show that it equals zero.

## 2. What is the significance of proving that Vector A and d\vec{}A/dt are perpendicular?

The significance of proving this is that it shows that the rate of change of Vector A is always perpendicular to Vector A itself. This is an important concept in vector calculus and has many applications in physics and engineering.

## 3. What is the constant magnitude in this proof?

The constant magnitude in this proof refers to the fact that the magnitude of Vector A remains constant, while its direction changes. This is important in proving that Vector A and d\vec{}A/dt are always perpendicular.

## 4. Can this proof be applied to any vector?

Yes, this proof can be applied to any vector, as long as its magnitude remains constant and it has a rate of change vector, d\vec{}A/dt. The proof relies on the dot product method, which can be applied to any two vectors.

## 5. What are some real-world examples of Vector A and d\vec{}A/dt being perpendicular with constant magnitude?

One example is a car driving in a circular motion. The velocity vector of the car, which is tangent to the circular path, is perpendicular to the acceleration vector, which is pointing towards the center of the circle. Another example is a swinging pendulum, where the velocity vector at any point is perpendicular to the acceleration vector due to gravity.

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