1. **Problem Statement:** We are given vectors $\mathbf{a}$ and $\mathbf{b}$ which are not parallel, and a vector $\overrightarrow{XY} = 2\mathbf{a} - 5\mathbf{b}$. We need to determine which of the given vectors are parallel to $\overrightarrow{XY}$. 2. **Key Concept:** Two vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel if there exists a scalar $k$ such that $\mathbf{u} = k\mathbf{v}$. 3. **Given vectors to check:** - $2\mathbf{a} - 5\mathbf{b} - 4\mathbf{a}$ - $-4\mathbf{a} + 10\mathbf{b}$ - $2\mathbf{a} + 5\mathbf{b}$ - $\mathbf{a} - \frac{5}{2}\mathbf{b}$ - $10\mathbf{a} - 25\mathbf{b}$ - $4\mathbf{a} - 20\mathbf{b} + 2\mathbf{a} + 5\mathbf{b}$ - $5\mathbf{a} - 2\mathbf{b}$ 4. **Simplify each vector:** - $2\mathbf{a} - 5\mathbf{b} - 4\mathbf{a} = (2 - 4)\mathbf{a} - 5\mathbf{b} = -2\mathbf{a} - 5\mathbf{b}$ - $-4\mathbf{a} + 10\mathbf{b}$ (already simplified) - $2\mathbf{a} + 5\mathbf{b}$ (already simplified) - $\mathbf{a} - \frac{5}{2}\mathbf{b}$ (already simplified) - $10\mathbf{a} - 25\mathbf{b}$ (already simplified) - $4\mathbf{a} - 20\mathbf{b} + 2\mathbf{a} + 5\mathbf{b} = (4 + 2)\mathbf{a} + (-20 + 5)\mathbf{b} = 6\mathbf{a} - 15\mathbf{b}$ - $5\mathbf{a} - 2\mathbf{b}$ (already simplified) 5. **Check for parallelism:** We want to see if each vector is a scalar multiple of $2\mathbf{a} - 5\mathbf{b}$. - For $-2\mathbf{a} - 5\mathbf{b}$: Compare coefficients with $2\mathbf{a} - 5\mathbf{b}$. The $\mathbf{a}$ coefficient is $-2$ vs $2$, ratio $-1$. The $\mathbf{b}$ coefficient is $-5$ vs $-5$, ratio $1$. Ratios differ, so not parallel. - For $-4\mathbf{a} + 10\mathbf{b}$: Coefficients compared to $2\mathbf{a} - 5\mathbf{b}$ are $-4$ vs $2$ (ratio $-2$) and $10$ vs $-5$ (ratio $-2$). Both ratios equal $-2$, so this vector is parallel. - For $2\mathbf{a} + 5\mathbf{b}$: Coefficients $2$ vs $2$ (ratio $1$), $5$ vs $-5$ (ratio $-1$). Ratios differ, not parallel. - For $\mathbf{a} - \frac{5}{2}\mathbf{b}$: Coefficients $1$ vs $2$ (ratio $\frac{1}{2}$), $-\frac{5}{2}$ vs $-5$ (ratio $\frac{1}{2}$). Both ratios equal $\frac{1}{2}$, so parallel. - For $10\mathbf{a} - 25\mathbf{b}$: Coefficients $10$ vs $2$ (ratio $5$), $-25$ vs $-5$ (ratio $5$). Both ratios equal $5$, so parallel. - For $6\mathbf{a} - 15\mathbf{b}$: Coefficients $6$ vs $2$ (ratio $3$), $-15$ vs $-5$ (ratio $3$). Both ratios equal $3$, so parallel. - For $5\mathbf{a} - 2\mathbf{b}$: Coefficients $5$ vs $2$ (ratio $2.5$), $-2$ vs $-5$ (ratio $0.4$). Ratios differ, not parallel. 6. **Final answer:** The vectors parallel to $\overrightarrow{XY} = 2\mathbf{a} - 5\mathbf{b}$ are: - $-4\mathbf{a} + 10\mathbf{b}$ - $\mathbf{a} - \frac{5}{2}\mathbf{b}$ - $10\mathbf{a} - 25\mathbf{b}$ - $6\mathbf{a} - 15\mathbf{b}$ (which is $4\mathbf{a} - 20\mathbf{b} + 2\mathbf{a} + 5\mathbf{b}$ simplified) These vectors are scalar multiples of $2\mathbf{a} - 5\mathbf{b}$ and thus parallel.