Scalar Triple Product Derivative

[SPhy]

Homework Statement

Find an expression equivalent for the derivative of the scalar triple product

a(t) . (b(t) x c(t))

The Attempt at a Solution

Initially I figured since whatever comes out of B X C is being dotted with A, I can use the derivative rules of a dot product:

(a(t)' . (b(t) x c(t))) + ( a(t) . (b(t) x c(t))' )

However, the solution just gives an expression for the scalar triple product in a 3x3 matrix form, that is,a(t) . (b(t) x c(t)) = [ a1 a2 a3
b1 b2 b3
c1 c2 c3 ]

What gives?

 

[LCKurtz]

Homework Statement

Find an expression equivalent for the derivative of the scalar triple product

a(t) . (b(t) x c(t))

The Attempt at a Solution

Initially I figured since whatever comes out of B X C is being dotted with A, I can use the derivative rules of a dot product:

(a(t)' . (b(t) x c(t))) + ( a(t) . (b(t) x c(t))' )

However, the solution just gives an expression for the scalar triple product in a 3x3 matrix form, that is,


a(t) . (b(t) x c(t)) = [ a1 a2 a3
b1 b2 b3
c1 c2 c3 ]

What gives?

That would be helpful if you know how to differentiate a determinant whose entries are functions. But I like your original idea. What is the formula for $(\vec b(t) \times \vec c(t))'$? Is there a similar formula as you used for the dot product? Just keep going...