Determine \( \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) \) if \( \mathbf{u}=\langle 3,-1\rangle, \mathbf{w}=\langle-2,4\rangle \), and \( \mathbf{u} \cdot \mathbf{v}=-5 \)

To determine \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \), we can use the distributive property of the dot product: \[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \] We are given: - \( \mathbf{u} = \langle 3, -1 \rangle \) - \( \mathbf{w} = \langle -2, 4 \rangle \) - \( \mathbf{u} \cdot \mathbf{v} = -5 \) First, we need to calculate \( \mathbf{u} \cdot \mathbf{w} \): \[ \mathbf{u} \cdot \mathbf{w} = \langle 3, -1 \rangle \cdot \langle -2, 4 \rangle \] Calculating the dot product: \[ \mathbf{u} \cdot \mathbf{w} = 3 \cdot (-2) + (-1) \cdot 4 = -6 - 4 = -10 \] Now we can substitute back into our equation: \[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = -5 + (-10) = -5 - 10 = -15 \] Thus, the final result is: \[ \boxed{-15} \]