Understand translations and their representation as column vectors.

A quadrilateral PQRS has vertices P(2, 3), Q(5, 7), R(8, 5), and S(4, 1). The quadrilateral undergoes a translation represented by the column vector \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\) to form a new quadrilateral P'Q'R'S'. a) Determine the coordinates of vertex R' after the translation. (1 mark) b) Express the translation from point S to point S' as a column vector. (1 mark) c) Show that the vector \(\overrightarrow{PQ}\) is equal to the vector \(\overrightarrow{P'Q'}\) by calculating both vectors. (1 mark)

Points \(A\) and \(B\) have position vectors \(\vec{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}\) and \(\vec{OB} = \begin{pmatrix} 5 \\ 3 \\ 1 \end{pmatrix}\) respectively, where \(O\) is the origin. a) Determine the translation vector \(\vec{AB}\) as a column vector. (1 mark) b) Point \(C\) is obtained by applying the translation \(\begin{pmatrix} -3 \\ 2 \\ 5 \end{pmatrix}\) to point \(B\). Determine the position vector of point \(C\). (1 mark) c) Show that the vector \(\vec{AC}\) can be expressed as the sum of \(\vec{AB}\) and the translation from \(B\) to \(C\). (1 mark)

Triangle \(ABC\) has vertices \(A(2, 3)\), \(B(5, 1)\), and \(C(4, 6)\). The triangle is translated by the vector \(\mathbf{t} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\) to form triangle \(A'B'C'\). a) Determine the coordinates of vertex \(A'\) after the translation. (1 mark) b) Determine the coordinates of vertices \(B'\) and \(C'\) after the translation. (1 mark) c) Show that the distance between \(A\) and \(B\) is equal to the distance between \(A'\) and \(B'\). (1 mark)