How can A be expressed in terms of n as a unit vector?

[aigerimzh]

Homework Statement

Let A be an arbitrary vector and let n be a unit vector in some fixed direction. Show that A=(A.n).n+(A*n)*n

Homework Equations

The Attempt at a Solution

I know that (A.n).n gives component of arbitrary vector, assume that it equals to A_x

 

[mathfeel]

Homework Statement

Let A be an arbitrary vector and let n be a unit vector in some fixed direction. Show that A=(A.n).n+(A*n)*n

Homework Equations

The Attempt at a Solution

I know that (A.n).n gives component of arbitrary vector, assume that it equals to A_x

Most straightforward way is to write out the Cartesian components and verify. Just keep in mind that $n_x^2 + n_y^2 + n_z^2 = 1$.

 

[HallsofIvy]

Again, you have used "*". What is that? The cross product? The usual notation is just "AX B".

 

[aigerimzh]

Yes, here also I mean (Axn)xn

 

[HallsofIvy]

You can set up you own coordinate system and so, without loss of generality, take n to be $\vec{i}$. Write A as $a\vec{i}+ b\vec{j}+ c\vec{c}$.

Then $A\cdot n= a$ so that $(A\cdot n)= a\vec{i}$. What are $A\times n$ and $(A\times n)\times n$?

 

[aigerimzh]

I think that (Axn)xn= aj?

 

[HallsofIvy]

I think that (Axn)xn= aj?

No. Try again. What is Axn first?