A 3×33 \times 33×3 matrix has its first two rows fixed and its third row unknown:
Q=13(2−21122abc)Q = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 1 & 2 & 2 \\ a & b & c \end{pmatrix}Q=3121a−22b12c
The scalars a,b,ca, b, ca,b,c are chosen so that QQQ is an orthogonal matrix (QTQ=IQ^{\mathsf{T}}Q = IQTQ=I) and detQ=+1\det Q = +1detQ=+1, i.e. QQQ is a pure rotation rather than a reflection.
What is (a, b, c)(a,\ b,\ c)(a, b, c)?
Select all that apply.
(−6, −3, 6)(-6,\ -3,\ 6)(−6, −3, 6)
(−2, 1, 2)(-2,\ 1,\ 2)(−2, 1, 2)
(−2, −1, 2)(-2,\ -1,\ 2)(−2, −1, 2)
(2, 1, −2)(2,\ 1,\ -2)(2, 1, −2)