- #1

Bashyboy

- 1,421

- 5

## Homework Statement

Let u be an arbitrary fixed unit vector and show that an vector b satisfies [itex]b^2 = (\vec{u} \cdot \vec{b}) + (\vec{u} \times \vec{b})^2[/itex] Explain this result in words, with the help of a picture.

## Homework Equations

## The Attempt at a Solution

I understand that the equations says that the square of the magnitude of some vector b is equal to the square of the dot product of b and some arbitrary unit vector u, plus the square of the cross product between the two vectors alluded to already.

I want to examine the dot product first. [itex]\vec{u} \cdot \vec{b} = |u||b|\cos \theta[/itex]. Is it correct to state that the cross product represents the amount of vector b that goes (points) in the direction of vector u. So, the right side of the equation can be thought of the magnitude of some vector[itex]\vec{b_{\vec{u}}}[/itex], such that [itex]\vec{b_{\vec{u}}} = c \vec{u}[/itex], and [itex]\vec{b} = \vec{b_{\vec{u}}} + \vec{b_{||}}[/itex], where [itex]\vec{b_{||}}[/itex] is orthogonal to the vector u.

Are these correct statements?