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

To simplify the expression \( 3 ( \vec { a } + \vec { b } ) - 4 ( 2 \vec { a } - \vec { b } ) + \vec { a } \), we can start by distributing the constants into the vectors. First, distribute the 3 and -4: \( 3\vec{a} + 3\vec{b} - (8\vec{a} - 4\vec{b}) + \vec{a} \) This simplifies to: \( 3\vec{a} + 3\vec{b} - 8\vec{a} + 4\vec{b} + \vec{a} \) Now, combine like terms: - For the \(\vec{a}\) terms: \(3\vec{a} - 8\vec{a} + \vec{a} = -4\vec{a}\) - For the \(\vec{b}\) terms: \(3\vec{b} + 4\vec{b} = 7\vec{b}\) Putting it all together, we get: \(-4\vec{a} + 7\vec{b}\) So the simplified expression is: \(-4\vec{a} + 7\vec{b}\)