Scalar Triple Product Derivative

In summary, the derivative of the scalar triple product a(t) . (b(t) x c(t)) can be expressed as the sum of the derivative of the first term and the derivative of the second term, using the derivative rules for a dot product. However, the solution may also be expressed as a 3x3 matrix composed of the entries a1, a2, a3, b1, b2, b3, c1, c2, c3. Further examination and use of the formula for the derivative of the cross product may lead to a simpler solution.
  • #1
SPhy
25
0

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?
 
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  • #2
SPhy said:

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...
 

Related to Scalar Triple Product Derivative

What is the definition of Scalar Triple Product Derivative?

The scalar triple product derivative is a mathematical operation that combines three vectors to produce a scalar value. It is calculated by taking the dot product of the cross product of two vectors with the third vector.

What is the geometric interpretation of Scalar Triple Product Derivative?

The geometric interpretation of the scalar triple product derivative is the volume of the parallelepiped formed by the three vectors. This can be visualized as the area of the base multiplied by the height.

How is Scalar Triple Product Derivative related to the cross product?

The scalar triple product derivative is directly related to the cross product, as it is calculated by taking the dot product of the cross product of two vectors with the third vector. This relationship can also be written as (a x b) ⋅ c.

What is the significance of Scalar Triple Product Derivative in physics?

The scalar triple product derivative is commonly used in physics to calculate the torque, or rotational force, on an object. It is also used in calculating the magnetic moment of a charged particle in a magnetic field.

What are some real-life applications of Scalar Triple Product Derivative?

The scalar triple product derivative has various real-life applications, including calculating the volume of a 3D object, determining the angle between two vectors, and calculating the work done by a force on an object in a certain direction.

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