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Let $\mathbf{a}$ be a vector of length $2$, and let $\mathbf{b}$ and $\mathbf{c}$ be vectors such that \[\mathbf{a} \bullet \mathbf{b} = 3, \mathbf{a} \bullet \mathbf{c} = 4, \mathbf{b} \bullet \mathbf{c} = 5.\]Enter in the numbers\[\mathbf{a}\bullet( \mathbf{b} + \mathbf{c}), \mathbf{b}\bullet(\mathbf{a} +\mathbf{b} + \mathbf{c}), (\mathbf{a} + 2\mathbf{b})\bullet(-3\mathbf{a} + 3\mathbf{c})\]below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter a question mark.
 May 14, 2023
 #1
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-1

The answers are 5, ?, -8.

 May 14, 2023

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