# Elementary physics question

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It's well known that the first derivative of the function which expresses the position of an object with respect to the time is the velocity of that object.

However the velocity is a vector and my question is - how can that be ?

Is this vector supposed to be constructed by the the absolute value of the derivative and the sign of the derivative ( which gives the direction ) or am i missing something ?

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Recall that a vector (for our purposes here) is a magnitude and a direction. The difference between speed and velocity here is that speed is analogous to the magnitude of the velocity vector.

Suppose that we're standing next to a highway that travels north and south. I could say that an object on the highway is traveling at 30 kilometers per hour without telling you which direction; this is describing only the speed of the object. I could also say that an object on the highway is traveling at 30 kilometers per hour headed south. Now that I've given both a magnitude (30) and a direction (south), I have described the velocity of the object.

I hope this helps.

Edited by 33andathirdRPM
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Take the derivative of x(t)i + y(t)j + z(t)k with respect to time. You'll notice that i, j, and k are unit vectors, in their respective dimensions. Since i, j, and k are not time dependent, they can be pulled out of the derivative. So according to the the addition rule you get:

d/dt(x(t)) * i + d/dt(y(t)) * j + d/dt(z(t)) *k, this is the velocity of the previous equation. As you can see, it has both a magnitude, which can be obtained from the scalar values in the equation and a direction, which can be obtained by the summation of the three terms itself, as vectors.

What perhaps many intro physics classes do, is, when first teaching the kinematic equations they take out the directional unit vector because things are only examined in one or two directions.

The method you suggest for finding a magnitudes direction is entirely false: There is no way to find direction from magnitude, or magnitude from direction. A vector, as hopefully you know consists of coordinates in how ever many dimensions the space you are operating in:

1 Dimension:

i = [1]

2 dimension:

i = [1,0]

3 dimension:

i = [1,0,0]

As you can see, all three vectors describe the same direction, but they are not all living in the same space (1 space , 2 space, 3 space). When working in 1 space, its sort of arbitrary to specify the vector, as the direction will be in either + or - 1 space. But once moving into 2 space, or planes as they are more commonly referred to, you have 360 degrees of rotation and thus it is important to specify the unit vectors and their corresponding magnitudes.

Let me know if you have any other questions.

Edited by GeoDUDE!
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GeoDUDE answered the question well with equations. I'll supplement the above explanation with some additional words.

A function is a mathematical construct where you put in some inputs and get an output. There are two types of functions: scalar functions and vector functions. With a scalar function, you get a scalar value out. Position is a vector function, so like GeoDUDE says, the output is a vector, not a scalar (the "input" for the position vector function is usually the time). When you take the first derivative of a vector function, you get another vector function, in this case, the velocity vector function.

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Thank you all for your time and knowledge.

Edited by hvasilev

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