Determine a unique vector x

[sigma]

Given vectors a and b, do the equations (x cross a = b) and (x dot a=the norm of a) determine a unique vector x? Argue both geometrically and analytically.

Not even sure where to begin with this question. If somebody could point me in the right direction, it would be much obliged!

 

[Opalg]

Here's a sketch of a geometric argument. First, x cross a has to be orthogonal to both x and a, so unless a and b are orthogonal the equations for x cannot have any solution at all.

Assuming that a and b are orthogonal, the equation x cross a = b tells you that x lies in the plane orthogonal to b. The equation x dot a = (norm of a) tells you that the component of x in the direction of a is 1. Finally the norm of (x cross a) tells you the size of the component of x in the direction orthogonal to a (in the plane containing x and a). These data together should determine x uniquely.