The point A A\,A has position vector (ab3)\begin{pmatrix} a \\ b \\ 3 \end{pmatrix}ab3, where a a\,a and b b\,b are positive constants, the point B B\,B has position vector (234)\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}234 and the point C C\,C has position vector (012)\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}012.
ABC ABC\,ABC is an isosceles triangle in which AB=ACAB = ACAB=AC.
Show that a+b=3a + b = 3a+b=3.
Given also that ∣AB⃗∣=6\left|\vec{AB}\right| = \sqrt{6}AB=6, find the exact values of a a\,a and bbb.