Differentiating Vector Products

[Identify]

Homework Statement

Prove that d/dt[r.(vxa)] = r.(vxda/dt)

Homework Equations

r, v, a are position, velocity and acceleration vectors.
..r.(v.. is the dot product.
..vxa.. is the cross product

The Attempt at a Solution

I expand the equation using the product rule for dot and cross products to get:

dr/dt.(vxa)+r.(dv/dt x a+v x da/dt)

I've expanded this further on paper using the x,y,z components of each vector but i can't manipulate it to get the desired result? Have I missed a step or overlooked something here?

 

[issacnewton]

well its way more simple than you are assuming. derivative of v with respect to time is a, so your second term is a cross product between a and a, so that becomes zero. In the first term, the time derivative of r is v. So it becomes
$\mathbf{v}\cdot (\mathbf{v}\times \mathbf{a})$ Use the property of vector triple product to make this zero. So you are left with the last term

 

[Identify]

Thanks very much.