Derivative of vector functions

In summary, the conversation is about someone struggling with a concept in vector calculus and seeking help. They are specifically trying to understand the derivative of a cross product and the correct formula for it. Through a series of back-and-forth questions and hints, it is determined that the answer is always 0 for the cross product of a vector with itself. The individual is grateful for the help and acknowledges their shaky understanding of vector calculus.
  • #1
LinearAlgebra
22
0
Okay, so I am having trouble with this concept (i actually didnt think i was until i realized i was getting all the wrong answers)...
As an example, if you want to take the derivative of:

d/dt [ r(t) X r'(t)]

I just used the general rule of d/dt [r1(t) X r2(t)] = r1(t)Xr2'(t) + r1'(t)Xr2(t)

I ***thought**** I was doing the problem correctly but apparently not.

I just did the following by using the generic equation above:
d/dt [ r(t) X r'(t)] = r(t)Xr''(t) + r'(t)Xr'(t)

What is wrong with what I did? At the back of my book it says the answer is r(t) Xr''(t) which is the first part of my answer (but doesn't include the second portion). Can someone explain where I'm wrong? I'd really appreciate it.
 
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  • #2
For any vector [itex]\vec a[/itex], what is [itex] \vec a \times \vec a[/itex]?
 
  • #3
the cross product?? the determinant of the components?

Sorry I'm really shaky on vector calculus...
 
  • #4
There is no vector calculus here. I asked what is [itex] \vec a \times \vec a[/itex]. Hint: The answer is the same for all vectors.

Step back: What is the magnitude of the cross product of two vectors [itex] \vec a[/itex] and [itex] \vec b [/itex]? What does that become when you set [itex] \vec b = \vec a[/itex]?
 
  • #5
a^2 ? so how does that help solve the problem?
 
  • #6
okay um nevermind i mean 0!
 
  • #7
Excellent. So now, what is

[tex]\vec r(t) \times \vec r\;^{\prime\prime}(t) + \vec r\;^\prime(t) \times \vec r\;^\prime(t)[/tex] ?
 

1. What is a vector function?

A vector function is a mathematical function that takes in one or more input variables and outputs a vector. It can be represented as a parametric equation, where each component of the vector is a function of the input variables.

2. What is the derivative of a vector function?

The derivative of a vector function is a vector that represents the rate of change of the function with respect to its input variables. It is calculated by taking the derivative of each component of the vector function separately.

3. How is the derivative of a vector function different from a scalar function?

The derivative of a scalar function is a single number, while the derivative of a vector function is a vector. This is because a scalar function only has one output, while a vector function has multiple outputs (components).

4. What is the geometric interpretation of the derivative of a vector function?

The derivative of a vector function can be interpreted as the tangent vector to the curve described by the function at a specific point. It represents the direction and rate of change of the curve at that point.

5. How is the derivative of a vector function used in physics?

The derivative of a vector function is used in physics to calculate velocity and acceleration of objects in motion. It is also used in vector calculus to solve problems related to forces, motion, and energy in three-dimensional space.

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