Determine a formula for (RE: vector valued functions)

In summary: The second part is r' x r'", so the second term is r.(r' x r'"). That's quite a bit simpler than your formula.In summary, the formula for d/dt [r.(r' x r'')] in terms of r is r'.(r'x r") + r.(r' x r'").
  • #1
WalkingInMud
5
0
Let r(t) be a v.v.f -with the first and second derivatives r' and r''. Determine formula for
d/dt [r.(r' x r'')] -in terms of r:

How do we approach this one?

maybe:
d/dt [r.(r' x r'')] = d/dt [r.r' x r.r''] = ((r'.r' + r.r'') x r'.r'') + (r.r' x (r'.r'' + r.r''')) ??
...and then what?

Is anyone able to give me a starting point -- or starting direction? -- thanks heaps
 
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  • #2
WalkingInMud said:
Let r(t) be a v.v.f -with the first and second derivatives r' and r''. Determine formula for
d/dt [r.(r' x r'')] -in terms of r:

How do we approach this one?

maybe:
d/dt [r.(r' x r'')] = d/dt [r.r' x r.r''] = ((r'.r' + r.r'') x r'.r'') + (r.r' x (r'.r'' + r.r''')) ??
...and then what?

Is anyone able to give me a starting point -- or starting direction? -- thanks heaps
What you have now doesn't make sense. r.r' is a number, not a vector. Same thing for r.r". You can't take the cross product. You seem to be assuming that u.(v x w)= (u.v)x(u.w) and, as I just said, that product doesn't make sense.

How about jusst using the product rule:
d/dt[r.(r' x r")]= r'.(r'x r")+ r.(r'+ r")'. The first term is easy: (r' x r") is perpendicular to both r' and r" so its dot product with r' is ?

Now expand (r' x r")' in the same way. Again, the first part of the sum is trivial.
 

1. What is a vector valued function?

A vector valued function is a mathematical function that takes in one or more input variables and returns a vector as the output. The vector is typically defined by its magnitude and direction, and can be represented as a list of coordinates.

2. How do you determine a formula for a vector valued function?

To determine a formula for a vector valued function, you need to identify the input variables and the components of the vector output. Then, you can use mathematical operations and functions to manipulate the input variables and create a formula that will produce the desired vector as the output.

3. What are some common examples of vector valued functions?

Some common examples of vector valued functions include parametric equations for curves and surfaces, velocity and acceleration functions in physics, and force and displacement functions in engineering. These functions are used to represent physical quantities that have both magnitude and direction.

4. How are vector valued functions different from scalar valued functions?

Vector valued functions differ from scalar valued functions in that they return a vector as the output instead of a single numerical value. Scalar valued functions only have one input variable, while vector valued functions may have multiple input variables to determine the components of the output vector.

5. What is the purpose of using vector valued functions?

Vector valued functions are used to represent physical quantities that have both magnitude and direction. They are also useful in modeling and analyzing complex systems, such as motion and forces in physics, where multiple variables and their interactions need to be considered. Vector valued functions are also essential in 3D graphics and computer animation, where they are used to create realistic and dynamic movements and shapes.

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