1. **Problem statement:** We have a parallelogram WXYZ with vectors \(\overrightarrow{WX} = \mathbf{u}\) and \(\overrightarrow{WZ} = \mathbf{v}\). M is the midpoint of \(\overrightarrow{WY}\). We need to express the following vectors in terms of \(\mathbf{u}\) and \(\mathbf{v}\): (i) \(\overrightarrow{WY}\), (ii) \(\overrightarrow{WM}\), (iii) \(\overrightarrow{XW}\), (iv) \(\overrightarrow{XZ}\). Then, show that M is the midpoint of \(\overrightarrow{XZ}\) using vectors. 2. **Recall properties of parallelograms and vectors:** - In a parallelogram, opposite sides are equal and parallel. - Vector addition rule: \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\). - Midpoint M of segment WY means \(\overrightarrow{WM} = \frac{1}{2} \overrightarrow{WY}\). 3. **Find each vector:** (i) \(\overrightarrow{WY}\): Since WXYZ is a parallelogram, \(\overrightarrow{WY} = \overrightarrow{WX} + \overrightarrow{XY}\). But \(\overrightarrow{XY} = \overrightarrow{WZ} = \mathbf{v}\) (opposite sides equal), so $$\overrightarrow{WY} = \mathbf{u} + \mathbf{v}.$$ (ii) \(\overrightarrow{WM}\): M is midpoint of WY, so $$\overrightarrow{WM} = \frac{1}{2} \overrightarrow{WY} = \frac{1}{2} (\mathbf{u} + \mathbf{v}).$$ (iii) \(\overrightarrow{XW}\): This is the vector from X to W, the opposite direction of \(\overrightarrow{WX}\), so $$\overrightarrow{XW} = -\mathbf{u}.$$ (iv) \(\overrightarrow{XZ}\): We can write \(\overrightarrow{XZ} = \overrightarrow{XW} + \overrightarrow{WZ} = -\mathbf{u} + \mathbf{v}.$$ 4. **Show M is midpoint of XZ:** We want to show that \(\overrightarrow{XM} = \frac{1}{2} \overrightarrow{XZ}\). Calculate \(\overrightarrow{XM}\): $$\overrightarrow{XM} = \overrightarrow{XW} + \overrightarrow{WM} = -\mathbf{u} + \frac{1}{2} (\mathbf{u} + \mathbf{v}) = -\mathbf{u} + \frac{1}{2} \mathbf{u} + \frac{1}{2} \mathbf{v} = -\frac{1}{2} \mathbf{u} + \frac{1}{2} \mathbf{v} = \frac{1}{2} (-\mathbf{u} + \mathbf{v}).$$ Recall from (iv) that \(\overrightarrow{XZ} = -\mathbf{u} + \mathbf{v}\), so $$\overrightarrow{XM} = \frac{1}{2} \overrightarrow{XZ}.$$ This confirms M is the midpoint of XZ. **Final answers:** (i) \(\overrightarrow{WY} = \mathbf{u} + \mathbf{v}\) (ii) \(\overrightarrow{WM} = \frac{1}{2} (\mathbf{u} + \mathbf{v})\) (iii) \(\overrightarrow{XW} = -\mathbf{u}\) (iv) \(\overrightarrow{XZ} = -\mathbf{u} + \mathbf{v}\) M is the midpoint of XZ because \(\overrightarrow{XM} = \frac{1}{2} \overrightarrow{XZ}\).