\( \vec { b } = 4 ( 2 \vec { u } - \vec { v } ) - 4 ( 3 \vec { u } - \vec { v } ) + 3 \vec { u } \)

To simplify \( \vec { b } = 4 ( 2 \vec { u } - \vec { v } ) - 4 ( 3 \vec { u } - \vec { v } ) + 3 \vec { u } \), you start by distributing the constants: 1. \( 4 ( 2 \vec { u } - \vec { v } ) = 8 \vec { u } - 4 \vec { v } \) 2. \( -4 ( 3 \vec { u } - \vec { v } ) = -12 \vec { u } + 4 \vec { v} \) Now, combine these results with the \( 3 \vec { u } \): \[ \vec { b } = (8 \vec { u } - 4 \vec { v}) + (-12 \vec { u} + 4 \vec { v}) + 3 \vec { u} \] Combine the \( \vec { u } \) terms and \( \vec { v } \) terms separately: \[ \vec { b } = (8 - 12 + 3) \vec { u} + (-4 + 4) \vec { v} = -1 \vec { u} + 0 \vec { v} = -\vec { u} \] So, \( \vec { b } = -\vec { u} \). This outcome shows that vector operations can sometimes boil down to simpler forms.