Proving Equation (1): Let r(t) be a Vector in $\mathbb{R^3}$

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In summary, the conversation discusses how to prove that the derivative of the vector product of the position vector and the cross product of the velocity and position vectors is equal to the norm of the position vector squared multiplied by the acceleration, plus the dot product of the position and velocity vectors multiplied by the velocity vector, minus the norm of the velocity vector squared plus the dot product of the position and acceleration vectors multiplied by the position vector. The conversation also mentions the use of formulas for the derivative of a vector product and the cross product of three vectors in $\mathbb{R^3}$. The speaker is seeking help in understanding how to apply these formulas to prove the given equation.
  • #1
WMDhamnekar
MHB
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Let r(t) be the position vector for a particle moving in $\mathbb{R^3}.$ How to show that
$$\frac{d}{dt}(r \times (v\times r))=||r||^2 *a+ (r\cdot v)*v-(||v||^2+ r\cdot a)*r \tag{1}$$

Where r(t) is a position vector (x(t),y(t),z(t)), $v(t)=\frac{dr}{dt}=(x'(t),y'(t),z'(t)), a(t)=\frac{dv}{dt}=\frac{d^2r}{dt^2}=(x''(t),y''(t),z''(t))$ Note: v=velocity, a= acceleration

My attempt:-
I know $\frac{d}{dt}(f\times g)=\frac{df}{dt}\times g +f\times \frac{dg}{dt}$. I also know for any vectors u, v, w in $\mathbb{R^3}, u\times (v\times w)=(u\cdot w)*v-(u\cdot v)*w $

But I don't understand how to use these formulas to prove equation (1)?

If any member knows answer to this question may reply with correct answer.
 
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  • #2
Hi Dhamnekar,

You're correct that those two formulas need to be used, nicely done. I will start the computation below. See if you can finish it. If not, I'm happy to answer any other questions you may have. Note: I will use the dot notation to denote a derivative with respect to time.

\begin{align*}

\frac{d}{dt}(r\times (v\times r)) &= \dot{r}\times (v\times r) + r\times \dot{(v\times r)}\\
& = v\times (v\times r) + r\times (\dot{v}\times r + v\times\dot{r})\\
& = v\times (v\times r) + r\times (a\times r + v\times v)

\end{align*}

Can you proceed from here? Hint: $x\cdot x = \| x\|^{2}.$
 
  • #3
\(\displaystyle \textbf{a} \times ( \textbf{b} \times \textbf{c} ) = ( \textbf{a} \cdot \textbf{c} ) \textbf{b} + ( \textbf{a} \cdot \textbf{b} ) \textbf{\)
Dhamnekar Winod said:
Let r(t) be the position vector for a particle moving in $\mathbb{R^3}.$ How to show that
$$\frac{d}{dt}(r \times (v\times r))=||r||^2 *a+ (r\cdot v)*v-(||v||^2+ r\cdot a)*r \tag{1}$$

Where r(t) is a position vector (x(t),y(t),z(t)), $v(t)=\frac{dr}{dt}=(x'(t),y'(t),z'(t)), a(t)=\frac{dv}{dt}=\frac{d^2r}{dt^2}=(x''(t),y''(t),z''(t))$ Note: v=velocity, a= acceleration

My attempt:-
I know $\frac{d}{dt}(f\times g)=\frac{df}{dt}\times g +f\times \frac{dg}{dt}$. I also know for any vectors u, v, w in $\mathbb{R^3}, u\times (v\times w)=(u\cdot w)*v-(u\cdot v)*w $

But I don't understand how to use these formulas to prove equation (1)?

If any member knows answer to this question may reply with correct answer.
Is this a part of a derivation? The \(\displaystyle \textbf{r} \times ( \textbf{v} \times \textbf{r} )\) looks familiar but I can't seem to find it.

-Dan
 

FAQ: Proving Equation (1): Let r(t) be a Vector in $\mathbb{R^3}$

1. What is the purpose of proving Equation (1)?

The purpose of proving Equation (1) is to demonstrate its validity and show that it accurately describes the relationship between the vector r(t) and the three-dimensional space it exists in. This proof is important in ensuring the accuracy and reliability of any further calculations or analysis that may involve this equation.

2. What does the notation r(t) represent?

The notation r(t) represents a vector in three-dimensional space, where t is a parameter that can take on different values. This vector can be thought of as a position vector, with its components representing the coordinates of a point in space at a given time t.

3. What is the significance of using vectors in $\mathbb{R^3}$?

Vectors in $\mathbb{R^3}$ are used to represent physical quantities that have both magnitude and direction in three-dimensional space. This is particularly useful in fields such as physics, engineering, and mathematics where three-dimensional problems are common.

4. How is Equation (1) proven?

Equation (1) is proven using mathematical techniques such as vector algebra and calculus. The proof involves manipulating the components of the vector r(t) and showing that they satisfy the conditions set forth in the equation. This may involve substitution, simplification, and other mathematical operations.

5. What are the implications of proving Equation (1)?

The implications of proving Equation (1) are that it can be used with confidence in further calculations and analysis. It also provides a deeper understanding of the relationship between r(t) and three-dimensional space, which can lead to new insights and discoveries in various scientific fields.

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