Finding Vectors for Cross Product Problem - Explanation and Solution

[JNBirDy]

Homework Statement

i) Find all vectors v such that <1,2,1> X v = <3,1,-5>

ii) Explain why there is no vector v such that <1,2,1> X v = <3,1,5>

Homework Equations

a X b = <a$_{2}$b$_{3}$ - a$_{3}$b$_{2}$, a$_{3}$b$_{1}$ - a$_{1}$b$_{3}$, a$_{1}$b$_{2}$ - a$_{2}$b$_{1}$)

The Attempt at a Solution

i)
<1,2,1> X v
= <2v$_{3}$ -v$_{2}$, v$_{1}$ - v$_{3}$, v$_{2}$ - 2v$_{1}$>

Which leaves me with three equations:
2v$_{3}$ -v$_{2}$ = 3
v$_{1}$ - v$_{3}$ = 1
v$_{2}$ - 2v$_{1}$ = -5

Now, every time I try to fiddle with them, I end up with 0 = 0.

Any help?

 

[ehild]

Express v2 from the first equation and v1 from the second one. v3 cancels when you substitute for v1 and v2 in the third equation, and you get -5=-5. This means that the three equations are not independent, and you have two equations for three variables: one variable is arbitrary.

ehild

 

[JNBirDy]

Express v2 from the first equation and v1 from the second one. v3 cancels when you substitute for v1 and v2 in the third equation, and you get -5=-5. This means that the three equations are not independent, and you have two equations for three variables: one variable is arbitrary.

ehild

Ah, yes, I think I get it now. For some reason I was thinking that I needed to get actual numbers.

Thanks.