taiwan-gsat 2023 Q5

taiwan-gsat · Other · gsat__math-a 5 marks Vector Product and Surfaces
It is known that $P$, $Q$, $R$ are three non-collinear points on the plane $2x - 3y + 5z = \sqrt{7}$ in coordinate space. Let $\overrightarrow{PQ} = (a_{1}, b_{1}, c_{1})$, $\overrightarrow{PR} = (a_{2}, b_{2}, c_{2})$. Select the option in which the absolute value of the determinant is the largest.
(1) $\left|\begin{array}{ccc} -1 & 1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(2) $\left|\begin{array}{ccc} 1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(3) $\left|\begin{array}{ccc} 1 & 1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(4) $\left|\begin{array}{ccc} -1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(5) $\left|\begin{array}{ccc} -1 & -1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
It is known that $P$, $Q$, $R$ are three non-collinear points on the plane $2x - 3y + 5z = \sqrt{7}$ in coordinate space. Let $\overrightarrow{PQ} = (a_{1}, b_{1}, c_{1})$, $\overrightarrow{PR} = (a_{2}, b_{2}, c_{2})$. Select the option in which the absolute value of the determinant is the largest.\\
(1) $\left|\begin{array}{ccc} -1 & 1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$\\
(2) $\left|\begin{array}{ccc} 1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$\\
(3) $\left|\begin{array}{ccc} 1 & 1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$\\
(4) $\left|\begin{array}{ccc} -1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$\\
(5) $\left|\begin{array}{ccc} -1 & -1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$