Linear Algebra Vectors (Find point P in R3 such that A,B,C and P form a rhombus)

[mmedy2311]

 

[stapel]

Consider the points $A = (1,\, 2,\, 3)$, $B = (-1,\, 4,\, 1)$, and $C = (3,\, 2,\, -2)$ in $\mathbb{R}^3$.

(a) Find the value of $k \in \mathbb{R}$ for which the vector $\boldsymbol{v} = [k,\, k - 2,\, 2k - 5]$ can be writen as a linear combination of $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

(b) Find the point $P$ in $\mathbb{R}^3$ such that $A$, $B$, $C$, and $P$ form a rhombus. Explain your reasoning.

What are your thoughts? What have you tried? How far have you gotten? ("Read Before Posting")

Please be complete, so the helpers can see where things are going sideways. Thank you!

 

[Steven G]

Have you found AB? How about AC?