1. **Problem Statement:** Given three non-coplanar vectors $a$, $b$, and $c$, express the cross products $b \times c$, $c \times a$, and $a \times b$ in terms of $a$, $b$, and $c$. 2. Because $a$, $b$, and $c$ are non-coplanar vectors, the set $\{a,b,c\}$ forms a basis in three-dimensional space. Therefore, each vector result of the cross product can be expressed as a linear combination of $a$, $b$, and $c$. 3. To express $b \times c$ in terms of $a$, $b$, and $c$, we use the formula involving the scalar triple product: $$b \times c = \frac{(a \cdot (b \times c))}{||a||^2} a + \text{(terms orthogonal to } a)$$ But more directly, the cyclic property of scalar triple product lets us write: $$b \times c = (a \times b) \times c = (a \cdot (b \times c)) \frac{a}{||a||^2}\quad \text{(if unit vectors or using scalar triple product)}$$ 4. However, the standard vector triple product identity helps here better: $$a \times (b \times c) = b(a \cdot c) - c(a \cdot b)$$ 5. Applying cyclic permutations, we can express each cross product as follows: $$b \times c = \alpha a$$ $$c \times a = \beta b$$ $$a \times b = \gamma c$$ where $\alpha = \frac{a \cdot (b \times c)}{||a||^2}$, and similarly for $\beta$ and $\gamma$. 6. Explicitly, in vector identity form: $$b \times c = (a \times b) \times c = (a \cdot (b \times c)) \frac{a}{||a||^2}$$ This shows that $b \times c$ is parallel to $a$ scaled by the scalar triple product. **Final Expressions:** $$b \times c = (a \cdot (b \times c)) \frac{a}{||a||^2}$$ $$c \times a = (b \cdot (c \times a)) \frac{b}{||b||^2}$$ $$a \times b = (c \cdot (a \times b)) \frac{c}{||c||^2}$$ These express the vector cross products in terms of $a$, $b$, and $c$ using scalar triple products and projections along the respective vectors.