\(F = q(v \times B)\)
\(F.B = 0\)
\(v \times B = \hat i [4B_0 - 6B] + \hat j [6B - 2B_0] + \hat k [2B - 4B]\)
\(= \hat i [4B_0 - 6B] + \hat j (6B- 2B_0)-\hat k(2B)\)
\(4B_0 - 6B = 4\) ......(1)
\(6B - 2B_0 = -20\) ......(2)
\(-2B = 12\) ......(3)
Solving (1), (2) and (3) we get
\(B = -6 \) and \(B_0 = - 8\)
\(\vec B = -6\hat i - 6\hat j - 8 \hat k\)