Proving Vector Orthogonality in R^3

[mjstyle]

Homework Statement

a) Let v be a unit vector in R^3 and u be a vector which is orthogonal to v. Show v x (v x u) = -u
b) Let v and u be orthogonal unit vectors in R^3. Show u x (v x (v x (v x u))) = -v

Homework Equations

The Attempt at a Solution

I am very lost in this question, I know a unit vector is = 1 therefore the summuation of the vector v is 1 for example, v = (1,0,0). square root(1^2 + 0 + 0) = 1 and i know u dot v is 0 but how do i start the prove?

thank you

 

[Defennder]

Apply the http://en.wikipedia.org/wiki/Triple_product" for the first one then simplify. Do it repeatedly for the second.

 

[matt grime]

That's one way to do it - the other is to think about the geometry. You know that u and v are orthogonal, so that u,v, vxu are all orthogonal, hence you know that vx(vxu) is parallel to u (since it is orthogonal to both v and vxu). What about its length? Again, just think about the geometric meaning. You should be able to show that vx(vxu) has length 1. Now you just need to consider if that means it is u or -u.

 

[HallsofIvy]

Since those equations are independent of choice of axes, it would be perfectly valid to choose you coordinate system so that u is pointing along the x-axis and v along the y-axis.

That is, assume that u= <a, 0, 0> and v= <0, b, 0> in the first problem and u= <1, 0, 0>, v= <0, 1, 0> in the other. Now use
$$\vec{u}\times\vec{v}= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ a & 0 & 0 \\0 & b & 0\end{array}\right|$$