Differentiating Vector v = ai + bxj: What Does it Mean?

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In summary: The magnitude of a vector is just its length. In this problem, a is just the magnitude of v - the vector is pointing towards (or towards the direction of) the positive x-axis.
  • #1
atavistic
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I have a vector v = a i +bxj

Now I want to find a.So I apply chain rule and differentiate dv/dx * dx/dt = bj * Vx = bj*ai ??

Now what does this mean? Is this allright? Should I not care about the vector directions when differentiating and just use the magnitudes a = ab , if yes then why? And what will be the direction of the acceleration then?
 
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  • #2
Hmm...

[tex]d/dt = \mathbf{v}\cdot \nabla[/tex]

[tex]
\frac{d\mathbf{v}}{dt} = \frac{\partial v_x}{\partial x}\frac{dx}{dt} + \frac{\partial v_y}{\partial y}\frac{dy}{dt}
[/tex]
and we know that

[tex]\frac{dx}{dt} = a, \frac{dy}{dt} = bx, \frac{\partial v_x}{\partial x} = 0, \frac{\partial v_y}{\partial y} = 0[/tex]

and then you get...
 
  • #3
I don't get what you have written. Please tell me the physics and not just the math. How can a be zero ?
 
Last edited:
  • #4
Hold on a second I did it wrong, didn't I? No one corrected me yet...

[tex]\frac{dv_x}{dt} = \frac{\partial v_x}{\partial x}\frac{dx}{dt} + \frac{\partial v_x}{\partial y}\frac{dy}{dt}[/tex]

[tex]\frac{dv_y}{dt} = \frac{\partial v_y}{\partial x}\frac{dx}{dt} + \frac{\partial v_y}{\partial y}\frac{dy}{dt}[/tex]

Which is just applying the chain rule to the components of v separately. Now that should make sense, it didn't before because it was wrong.
 
  • #5
[tex]a_x = 0[/tex]

[tex]a_y = ab[/tex]

I'm sorry about my stupidity.
 
  • #6
just differentiate the magnitudes. derivatives does not effect the basis vectors. so you write the related basis vector after the derivation. and it will give you the direction of the acceleration.
 
  • #7
torehan said:
just differentiate the magnitudes.

That's wrong. Consider the counterexample-- centripetal motion. The speed is constant in time, so by your method you would conclude that acceleration is 0, but it's not.

And in this problem, differentiating the magnitude would also give you the wrong answer.
 
  • #8
that's not totally wrong. if you should use "the polar coordinate" system and then you can find the direction.
 
  • #9
torehan said:
that's not totally wrong. if you should use "the polar coordinate" system and then you can find the direction.

No, the polar coordinate basis vectors are not constant.

Face it, once you leave 1 dimensional motion, you can not characterize position, velocity and acceleration by their magnitudes.
 
  • #10
atavistic said:
I have a vector v = a i +bxj

Now I want to find a.So I apply chain rule and differentiate dv/dx * dx/dt = bj * Vx = bj*ai ??

Now what does this mean? Is this allright? Should I not care about the vector directions when differentiating and just use the magnitudes a = ab , if yes then why? And what will be the direction of the acceleration then?

I'm not sure I understand, is a a constant? what do you understand by the vector a ?
 

1. What is a vector?

A vector is a mathematical object that has both magnitude and direction. It is often represented as an arrow pointing in a specific direction, with the length of the arrow indicating the magnitude.

2. What is the difference between a scalar and a vector?

A scalar is a single value, while a vector has both magnitude and direction. For example, temperature is a scalar quantity, while velocity is a vector quantity.

3. What is the purpose of differentiating a vector?

Differentiating a vector allows us to find the rate of change of the vector with respect to a specific variable. This can help us understand how the vector is changing over time or in different conditions.

4. What do the terms "ai" and "bxj" represent in the equation?

These terms represent the components of the vector in the x and y directions, respectively. "a" and "b" are scalar values that represent the magnitude of the vector in each direction, while "i" and "j" are unit vectors that indicate the direction.

5. How can I use the equation to solve a problem?

The equation allows you to break down a vector into its individual components and then use calculus to find the rate of change of each component. This can be useful in many fields, such as physics, engineering, and economics.

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