Given the vectors:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]
where
\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]
\[ \mathbf{a} \cdot \mathbf{x} = 3 \]
Find:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]
\( 36 \)
Vector Problem
Step 1: Given Vectors
We have:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}, \quad \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}. \]
Also, it is given that:
\[ \mathbf{a} \times \mathbf{c} = \mathbf{b}. \]
Step 2: Computing the Expression
We need to evaluate:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}). \]
Using the vector identity:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b}) = (\mathbf{a} \times \mathbf{x}) \cdot \mathbf{b}. \]
Since it is given that \( \mathbf{a} \cdot \mathbf{x} = 3 \), we compute:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b}) = (\mathbf{a} \times \mathbf{x}) \cdot \mathbf{b} = 3 (\mathbf{b} \cdot \mathbf{b}). \]
Computing \( \mathbf{b} \cdot \mathbf{b} \):
\[ (3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}) \cdot (3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}) = 9 + 9 + 9 = 27. \]
Thus,
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b}) = 3 \times 27 = 81. \]
Since \( \mathbf{a} \times \mathbf{c} = \mathbf{b} \), we have:
\[ \mathbf{a} \cdot (-\mathbf{c}) = -\mathbf{a} \cdot \mathbf{c} = -\mathbf{a} \cdot \mathbf{b}. \]
From the given condition, we know:
\[ \mathbf{a} \cdot \mathbf{b} = 57. \]
Thus,
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) = 81 - 57 = 24. \]
Step 3: Conclusion
Thus, the final answer is:
\[ \boxed{24}. \]
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\[ |\vec{a} + \vec{b} + \vec{c}| = \ ? \]
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