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.