Locus of points

**free_to_fly** · October 18th 2008, 02:59 PM

Q: Describe the locus of points that satisfy the equation:
$ r \wedge a = b $
where $ \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.

**mr fantastic** · October 18th 2008, 03:30 PM

Originally Posted by free_to_fly

Q: Describe the locus of points that satisfy the equation:
$ r \wedge a = b $
where $ \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 $r = <\alpha, \, \beta, \, \gamma>$ .

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

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

This is the vector equation of a line.

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