# Locus of points

• Oct 18th 2008, 02:59 PM
free_to_fly
Locus of points
Q: Describe the locus of points that satisfy the equation:
$\displaystyle r \wedge a = b$
where $\displaystyle \begin{array}{l} a = (1,1,0) \\ b = (1, - 1,0) \\ \end{array}$.

My thinking was that the locus is either a line or a plane, where b is perpendicular to a, but I'm not entirely sure which one it is. Help would be loved.
• Oct 18th 2008, 03:30 PM
mr fantastic
Quote:

Originally Posted by free_to_fly
Q: Describe the locus of points that satisfy the equation:
$\displaystyle r \wedge a = b$
where $\displaystyle \begin{array}{l} a = (1,1,0) \\ b = (1, - 1,0) \\ \end{array}$.

My thinking was that the locus is either a line or a plane, where b is perpendicular to a, but I'm not entirely sure which one it is. Help would be loved.

There are probably better ways than this:

Let $\displaystyle r = <\alpha, \, \beta, \, \gamma>$.

Then $\displaystyle r \wedge a = <-\gamma, \, \gamma, \, (\alpha - \beta)>$.

Therefore $\displaystyle r = <\alpha, \, \alpha, \, -1> = <0, \, 0, \, -1> + \alpha <1, 1, 0>$ for all real values of $\displaystyle \alpha$.

This is the vector equation of a line.