Question

Searched google for a half hour and couldn't find anything that helpful. The best I could find was http://www.math.jhu.edu/~vlorman/32.pdf


Given vectors a and b, do teh equations x CROSS a = b and x DOT a = ||a|| determine a unique vector x? Argue both geometrically and analytically.

Any help is appreciated.

Answer 1

i think we can use
x dot a = |x| |a| cos A
and
| x cross a | = |x| |a| sin A
to show we get x to a sign (i.e. two possible directions)

Answer 2

if vectors a and b are not orthogonal, then there is no x such that
x cross a = b (because b must be orthogonal to both x and a)

if a and b are not zero and orthogonal, we get a unique x
x will form an angle A= atan( |b|/|a|) with vector a, and have length= |a| tan A