Multi-variable calculus orhogonality problem

[K29]

Homework Statement

The curve $$p:R->R^{n}$$ and the vector $$v \in R^{n}$$. Assume $$v$$ and $$p'(t)$$ are orthogonal for all $$t$$. And that $$p(0)$$ is orthogonal to $$v$$.
Prove that $$p(t)$$ and $$v$$ are orthogonal for all t.

Homework Equations

Since the previous question in the same main question (ie 2(a) and we are now working on 2(b)is the following proof who's result may be relevant.
$$(u\cdot v)' = u\cdot v' +v\cdot u'$$
$$u,v\in R^{n}$$

The Attempt at a Solution

I can see that this should be true, just struggling with how to prove it:
I've applied the above relation to both the given dot products and set them equal to zero, since they are orthogonal. I've also expanded
$$(p(t) \cdot v)'$$ with the same above relation and I end up with 4 equations which may have been useful:
$$v\cdot p'(t)=0$$
$$v\cdot p(0)=0$$
$$(v\cdot p(0))'=v\cdot p(0)'+p(0)\cdot v'=0$$
$$(v\cdot p(t))'=v\cdot p'(t)+p(t)\cdot v'$$
however I see no way to relate u.v itself(rather than the derivative) using this method. I've substituted the above 4 equations into each other in various ways, however there is a fundamental relationship that I think I am missing that somehow links p(0) to everything else.
I've also tried writing the last two equations as summantions(dot product definition); and as the full matrix that the derivatives would yield, but the result is much the same dead end, other than the fact that I can pull the derivative out of the summations on the left hand side of the equations

Help pls?

 

[CompuChip]

Isn't this much easier than it looks?
Applying the identity for u = p(t), v = v gives you
$$(p(t) \cdot v)' = p'(t) \cdot v + p(t) \cdot v'$$

What does it mean for two vectors to be orthogonal?
Now what can you say about $p'(t) \cdot v$ and about v'?

 

[K29]

If they are orthogonal the dot product is zero, because the vectors are sitting perpendicularly at 90 degrees and uvcos(90)=0

We know that $$p'(t)\cdot v$$ is zero.
This leaves:
$$(p(t) \cdot v)' = p(t) \cdot v'$$

Unfortunately I don't see anything specific about v' other than it's perpendicular only to p(0) which was given..

 

[CompuChip]

Isn't v a constant vector?

 

[K29]

nvm, solved. Thank U for the constant vector clue, I missed that.. :), .I have difficulty seeing the subtleness of some proofs.