1. **State the problem:** We need to find which vectors from the given list are parallel to the vector $$\overrightarrow{FG} = 2a - 3b$$ where $$a$$ and $$b$$ are non-parallel vectors. 2. **Recall the rule for parallel vectors:** Two vectors $$\vec{u}$$ and $$\vec{v}$$ are parallel if there exists a scalar $$k$$ such that $$\vec{u} = k \vec{v}$$. 3. **Check each vector:** We want to see if each vector can be written as $$k(2a - 3b)$$ for some scalar $$k$$. - Vector 1: $$a - \frac{3}{2}b$$ Compare to $$k(2a - 3b) = 2k a - 3k b$$. Equate coefficients: $$1 = 2k \Rightarrow k = \frac{1}{2}$$ $$-\frac{3}{2} = -3k \Rightarrow k = \frac{1}{2}$$ Both match, so this vector is parallel. - Vector 2: $$10a - 15b$$ Equate coefficients: $$10 = 2k \Rightarrow k = 5$$ $$-15 = -3k \Rightarrow k = 5$$ Both match, so this vector is parallel. - Vector 3: $$3a - 2b$$ Equate coefficients: $$3 = 2k \Rightarrow k = \frac{3}{2}$$ $$-2 = -3k \Rightarrow k = \frac{2}{3}$$ Values of $$k$$ do not match, so not parallel. - Vector 4: $$-4a + 6b$$ Equate coefficients: $$-4 = 2k \Rightarrow k = -2$$ $$6 = -3k \Rightarrow k = -2$$ Both match, so this vector is parallel. - Vector 5: $$2a + 3b$$ Equate coefficients: $$2 = 2k \Rightarrow k = 1$$ $$3 = -3k \Rightarrow k = -1$$ Values of $$k$$ do not match, so not parallel. - Vector 6: $$2a - 3b - 4a$$ simplifies to $$-2a - 3b$$ Equate coefficients: $$-2 = 2k \Rightarrow k = -1$$ $$-3 = -3k \Rightarrow k = 1$$ Values of $$k$$ do not match, so not parallel. - Vector 7: $$2a - 12b + 4a + 3b$$ simplifies to $$6a - 9b$$ Equate coefficients: $$6 = 2k \Rightarrow k = 3$$ $$-9 = -3k \Rightarrow k = 3$$ Both match, so this vector is parallel. **Final answer:** The vectors parallel to $$\overrightarrow{FG}$$ are: $$a - \frac{3}{2}b, \quad 10a - 15b, \quad -4a + 6b, \quad 6a - 9b$$