But have you differentiated?karens said:Homework Statement
n.
Let r1 and r2 be differentiable 3-space vector-valued functions.
Directly from the definition of dot product, and the definition of derivative of vector-
valued functions in terms of components, prove that
d/dt (r1(t) • r2(t)) = r′1(t) • r2(t) + r1(t) • r′2(t).
Homework Equations
Dot Product: U • V = u1v1 + u2v2 + u3v3
The Attempt at a Solution
I can just substitute things forever and get nowhere...
A vector derivative is a mathematical operation that calculates the instantaneous rate of change of a vector with respect to a variable. It is similar to a regular derivative, but takes into account the direction of the vector as well as its magnitude.
The dot product is used because it is a mathematical operation that calculates the projection of one vector onto another. This is essential in determining the rate of change of a vector, as it allows us to measure the change in the direction of the vector.
The steps to prove the vector derivative using dot product are as follows:
1. Write the vector as a function of the variable
2. Use the definition of the dot product to expand the expression
3. Apply the limit definition of the derivative
4. Simplify the expression and factor out the variable
5. Use the properties of the dot product to rearrange the terms
6. Take the limit as the variable approaches zero
7. The resulting expression is the vector derivative.
The gradient is a vector that contains the partial derivatives of a multivariable function. The vector derivative is also a vector that contains the derivatives of a vector function. They are related in that the gradient is the vector derivative of a scalar function, while the vector derivative is the gradient of a vector function.
The vector derivative is used in physics, engineering, and other fields to model the motion and behavior of objects. It is used to calculate velocity, acceleration, and other important quantities that describe the movement of objects in space.