1. **State the problem:** We have three shapes X, Y, and Z. Shape X is translated by vector $$\begin{pmatrix} -2 \\ 5 \end{pmatrix}$$ to get shape Y. 2. Shape Y is then translated by vector $$\begin{pmatrix} 7 \\ -3 \end{pmatrix}$$ to get shape Z. 3. We want to find: a) The translation vector from shape X to shape Z. b) The translation vector from shape Z to shape X. 4. **Find translation from X to Z:** The total translation from X to Z is the sum of the two translations: $$\begin{pmatrix} -2 \\ 5 \end{pmatrix} + \begin{pmatrix} 7 \\ -3 \end{pmatrix} = \begin{pmatrix} -2 + 7 \\ 5 + (-3) \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$$ 5. **Find translation from Z to X:** This is the inverse of the translation from X to Z, so we negate the vector: $$-\begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} -5 \\ -2 \end{pmatrix}$$ **Final answers:** - a) Translation from X to Z is $$\begin{pmatrix} 5 \\ 2 \end{pmatrix}$$. - b) Translation from Z to X is $$\begin{pmatrix} -5 \\ -2 \end{pmatrix}$$.