Express the following vectors in terms of z and x. Simplify your answers where necessary. a) BC = BA + AC, b) BM = 1/2 BC

The problem states the vector sum:

$$\vec{BC} = \vec{BA} + \vec{AC}$$

From the student's work, it is implied that $\vec{BA} = -z$ and $\vec{AC} = x$. Substitute these into the equation:

$$\vec{BC} = (-z) + (x)$$

$$\vec{BC} = x - z$$

The simplified expression for $\vec{BC}$ is:

$$\vec{BC} = \boxed{x-z}$$

The problem states the relationship for $\vec{BM}$:

$$\vec{BM} = \frac{1}{2} \vec{BC}$$

From part a), we found that $\vec{BC} = x - z$. Substitute this into the equation for $\vec{BM}$:

$$\vec{BM} = \frac{1}{2} (x - z)$$

$$\vec{BM} = \frac{x - z}{2}$$

The simplified expression for $\vec{BM}$ is:

$$\vec{BM} = \boxed{\frac{x-z}{2}}$$