# Are Orthogonal Vectors Proven by Derivative and Dot Product?

• guyvsdcsniper
In summary: By taking the derivative of a constant vector with constant magnitude, we get a zero vector. When we dot product this with the original vector, the result is also zero. And when two vectors have a dot product of zero, they are orthogonal. Therefore, v'(t) is orthogonal to v(t).
guyvsdcsniper
Homework Statement
Prove that if v(t) is any vector that depends on time, but v(t) has constant magnitude, then
v˙(t) is orthogonal to v(t)
Relevant Equations
Dot Product
I feel like this question is very straight forward and my explanation below summarizes the answer pretty well. Could someone confirm this or tell me if I am missing something?

We have V which is a vector, but the question states it is a constant. If I take the derivative of V, represented by V', a constant, then I get 0.
If I dot product these to values, the product is then 0. And it is known that when the dot product between two vectors is zero, they are orthogonal.

quittingthecult said:
Homework Statement:: Prove that if v(t) is any vector that depends on time, but v(t) has constant magnitude, then
v˙(t) is orthogonal to v(t)
Relevant Equations:: Dot Product

I feel like this question is very straight forward and my explanation below summarizes the answer pretty well. Could someone confirm this or tell me if I am missing something?

We have V which is a vector, but the question states it is a constant. If I take the derivative of V, represented by V', a constant, then I get 0.
If I dot product these to values, the product is then 0. And it is known that when the dot product between two vectors is zero, they are orthogonal.
It says that ##\vec v## has constant magnitude; not that ##\vec v## is constant.

Orodruin
PeroK said:
It says that ##\vec v## has constant magnitude; not that ##\vec v## is constant.
Ah I misread the question. That makes a lot of sense. Thank you for catching that.

So what does it look like now?

BvU said:
So what does it look like now?
This is the conclusion I came to.

SammyS, Orodruin, BvU and 1 other person

## 1. What does it mean for vectors to be orthogonal?

Orthogonal vectors are those that are perpendicular to each other, meaning they form a 90-degree angle. This means that their dot product is equal to zero.

## 2. How do you prove that vectors are orthogonal?

To prove that two vectors are orthogonal, you can use the dot product formula: A · B = |A| * |B| * cos(θ), where θ is the angle between the two vectors. If the dot product is equal to zero, then the vectors are orthogonal.

## 3. Can two non-zero vectors be orthogonal?

Yes, two non-zero vectors can be orthogonal if their dot product is equal to zero. This means that they are perpendicular to each other, even if they have different magnitudes.

## 4. How can you visually determine if vectors are orthogonal?

You can visually determine if vectors are orthogonal by graphing them on a coordinate plane. If the vectors form a 90-degree angle, then they are orthogonal.

## 5. Are orthogonal vectors always linearly independent?

Yes, orthogonal vectors are always linearly independent. This means that they cannot be written as a linear combination of each other, and they are not dependent on each other's values.

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