Prove Vector Derivative (using Dot Product)

[karens]

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...

 

[lanedance]

show us how?

in this example the dot product is a scalar valued function of t, use the component form you have shown & the product rule and it should follow straight forward

 

[HallsofIvy]

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...

But have you differentiated?

Since, as you write, U• V= u1v1+ u2v2+ u3v3, (U• V)'= u1'v1+ u1v1'+ u2'v2+ u2v2'+ u3'v3+ u3v3'= (u1'v1+ u2'v2+ u3'v3)+ (u1v1'+ u2v2'+ u3v3').