# Velocity is a vector in Newtonian mechanics

• BookWei
In summary, the vector analysis in Arfken and Weber's textbook, Mathematical Methods for Physicists 5th edition, defines vectors in N dimensions as a set of N quantities, with their values relative to rotated coordinate axes given by a specific formula. The book also explains how this method can be used to prove that velocity is a vector in Newtonian mechanics.
BookWei
I studied the vector analysis in Arfken and Weber's textbook : Mathematical Methods for Physicists 5th edition.
In this book they give the definition of vectors in N dimensions as the following:
The set of ##N## quantities ##V_{j}## is said to be the components of an N-dimensional vector ##V##
if and only if their values relative to the rotated coordinate axes are given by
$$V_{i}^{'}=\sum_{j=1}^N a_{ij}V_{j},\;i=1,2,...,N$$
From the definition of ##a_{ij}## as the cosine of the angle between the positive ##x_{i}^{'}## direction
and the positive ##x_{j}## direction we may write (Cartesian coordinates)
$$a_{ij}=\frac {\partial x_{i}^{'}} {\partial x_{j}}$$

Can we use the same mathematical method (or tensor analysis) to prove the velocity is a vector in
Newtonian mechanics?
Many thanks!

BookWei said:
Can we use the same mathematical method (or tensor analysis) to prove the velocity is a vector in
Newtonian mechanics?
Why not? If you have a position vector ##\vec X## that transforms according to ##X_{i}^{'}=\sum^3_{j=1} a_{ij}X_{j}##, then when you take the time derivative, ##\dot{X}_{i}^{'}=\sum_{j=1}^3 a_{ij}\dot{X}_{j}##. This says that ##\dot{\vec X}## transforms as a vector.

## What is velocity in Newtonian mechanics?

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It includes both the magnitude and direction of an object's motion.

## How is velocity different from speed in Newtonian mechanics?

While both velocity and speed describe an object's motion, velocity also includes the direction of motion, while speed does not. This means that two objects can have the same speed but different velocities if they are moving in different directions.

## How is velocity calculated in Newtonian mechanics?

In Newtonian mechanics, velocity is calculated by dividing the change in an object's displacement by the change in time. This can be represented by the equation v = Δx/Δt, where v is velocity, Δx is change in displacement, and Δt is change in time.

## What is the SI unit for velocity in Newtonian mechanics?

The SI unit for velocity in Newtonian mechanics is meters per second (m/s). This unit represents the distance an object travels in meters divided by the time it takes to travel that distance in seconds.

## Can velocity be negative in Newtonian mechanics?

Yes, velocity can be negative in Newtonian mechanics. A negative velocity indicates that an object is moving in the opposite direction of a chosen reference point. For example, if a car is moving east but its velocity is given as -20 m/s, this means that it is moving west relative to a reference point.

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