Proof that a parametrized curve orthogonal to a certain vector v

Let be a parametrized curve and let be a fixed vector. Assume that is orthogonal to

for all and that is also orthogonal to

Prove that is orthogonal to for all

We are given that and is orthogonal to

Let and

So we have:

Now how do I use this info and prove that is orthogonal to for all ?

Is it possible for anyone to kindly give some hints on how to prove this?

Re: Proof that a parametrized curve orthogonal to a certain vector v

I think I solved it. For those who are interested:

Since

Particularly:

Taking integration of both sides:

By taking all constants to the right side we get:

We also know that:

So pluging in into eq(1) we get:

therefore:

But this means:

So is orthogonal to .[As was to be shown]

Re: Proof that a parametrized curve orthogonal to a certain vector v

Looks good but I think I would go the "other way"- that is, differentiate rather than integrate .

Let . Then, since v is a constant vector, . Since the derivative of u is 0, u is a constant. And since it follows that for all t.

Re: Proof that a parametrized curve orthogonal to a certain vector v

Quote:

Originally Posted by

**HallsofIvy** Looks good but I think I would go the "other way"- that is, differentiate

rather than integrate

.

Let

. Then, since v is a constant vector,

. Since the derivative of u is 0, u is a constant. And since

it follows that

for all t.

Thank you for showing another way of solving this problem. I believe your way is the correct and better method. Again thanks.