Finding B Using Cross Product in Magnetic Force Equation

[proton]

Homework Statement

find B from F=q(v X B), where F is magnetic force, q = charge, v = velocity, B = magnetic field.

Carrying out 3 experiments, we find that if
v_1 = i, (F/q)_1 = 2k - 4j
v_2 = j, (F/q)_2 = 4i - k
v_3 = k, (F/q)_3 = j - 2i

where i,j,k are the unit cartesian vectors

This is the problem 1.4.16 from Arfken's Mathematical methods for physicists

The Attempt at a Solution

I tried adding the v's and F's as follows:
[(v_1 X B)+ (v_2 X B) +(v_3 X B)] = - [(B X v_1)+ (B X v_2) +(B X v_3)] = -[B X (v_1 + v_2 + v_3)] = [(F/q)_1 + (F/q)_2 + (F/q)_3]
=> -[B X (i + j + k)] = [(2i - 4j) + (4i - k) + (j - 2i)] = 2i - 3j + k
=> [B X (i + j + k)] = -2i + 3j - k

multiplying out the cross product, I got: [B X (i + j + k)] = (B_y - B_z)i - (B_x - B_z)j + (B_x - B_y)i
=> B_y - B_z = -2
B_x - B_z = -3
B_x - B_y = -1

and this gives infinitely many solns for B_x, B_y, and B_z

is this correct? or did I screw up somewhere?

 

[D H]

Carrying out 3 experiments, we find that if
v_1 = i, (F/q)_1 = 2i - 4j

Since v_1 = i, how can F_1 = q(v_1xB) have any component in the i direction?

 

[proton]

typo on my part

 

[proton]

come on, can't someone help me?