Consider the points [imath]A = (1,\, 2,\, 3)[/imath], [imath]B = (-1,\, 4,\, 1)[/imath], and [imath]C = (3,\, 2,\, -2)[/imath] in [imath]\mathbb{R}^3[/imath].
(a) Find the value of [imath]k \in \mathbb{R}[/imath] for which the vector [imath]\boldsymbol{v} = [k,\, k - 2,\, 2k - 5][/imath] can be writen as a linear combination of [imath]\overrightarrow{AB}[/imath] and [imath]\overrightarrow{AC}[/imath].
(b) Find the point [imath]P[/imath] in [imath]\mathbb{R}^3[/imath] such that [imath]A[/imath], [imath]B[/imath], [imath]C[/imath], and [imath]P[/imath] form a rhombus. Explain your reasoning.