Need Help with Mathematical Proof? - Introducing Myself & Seeking Assistance

[advphys]

Dear all,
I am new to this forum and this is my first post.
I hope this is the right place to open this topic.

I have a homework for my mathematical methods for physics course and i am kind of stuck into a question:

v=ψ∇β
show that ∇xv is perpendicular to v.

Here v is a vector
x is cross product
∇ is gradient vector operator
ψ and β are non-vectoral functions

I know it seems obvious, but i need mathematical proof which probably contains some identities of triple cross product with del operator.
Like i said, i couldn't figure it out.

Already now, i want to thank you for all your help.

 

[lurflurf]

hint
∇xv=(∇ψ)x(∇β)
which is obvious from
∇x∇β=0

 

[jfgobin]

Hello AdvPhys,a) Use the identity of the curl of a scalar function multiplying a vector function
b) Use the fact that the curl of the gradient of a scalar function is 0
c) How do you prove that two vectors are perpendicular?
d) Conclude.

[Edit: previous post was giving too much info]

 

[advphys]

Thanks for replies.
But still can't figure out how to conclude the last scalar triple product gives 0.

Edit: I got that it is obvious. I have the same vector in both parts.

Again thank you for your help.