> [!summary] Vector equations are linear combinations of vectors and coefficients in a set, that result in a vector > **Key Equations:** $\vec{x} = \{ c_1 \vec{v_1} + \dots + c_k \vec{v_k} \space | c_1 \dots c_k \in \mathbb{R} \}$ >[!info]+ Read Time **⏱ 1 min** # Definition Vector equations are [[Linear Combinations|linear combinations]] of [[Vector Addition & Scalar Multiplication|vectors]] in a [[Sets|set]] ($\{ \vec{v_1} + \dots + \vec{v_k} \}$) that result in a new vector ($\vec{x}$). Since you can reach a vector ($\vec{x}$) using combinations of vectors and coefficients the formal definition is as followed $ \vec{x} = \{ c_1 \vec{v_1} + \dots + c_k \vec{v_k} \space | c_1 \dots c_k \in \mathbb{R} \} $ > [!note] Why does this equation work? Suppose I can reach $\vec{x}$ by combining two vectors $\vec{a}+\vec{b}$. But I can also reach $\vec{x}$ by combining two vectors $c_{1}\vec{v}+c_{2} \vec{v_{2}}$ where $c_{1},c_{1}$ are [[Scalar & Vectors|scalars]]. > The vector equation generalizes this result.