What are the steps to solving this multivariable calculus proof?

[sh3lF1sh]

Let a; b and c be any three vectors in R3 with a =6 0. Show that b = c if and only if
a
dot b = a dot
c and a cross b = a cross c

 

[lineintegral1]

To prove an if and only if statement, you must prove both directions of the theorem. In this case, you must show that [itex]b=c[/itex] implies the dot and cross product equations you provided. You must also show that the dot and cross product equations imply that [itex]b=c[/itex] (which is certainly more challenging, the first direction is relatively trivial).

To prove the latter statement I described, think about the definitions of the cross and dot product. Yes, you've been provided an easy means of calculating them (i.e. [itex]<x,y>\cdot <a,b>=ax+by[/itex]), but there is a trigonometric definition for the dot and cross products. Can you manipulate these to show that a=b? Hint: Consider the angle a makes between b and c individually. What must be true about these angles?