Question

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:

\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]

• A. \( 32 \)
• B. \( 24 \)
• C. \( 20 \)
• D. \( 36 \)

Hint:

Use vector triple product identities and scalar products to simplify expressions efficiently. Recognizing given conditions helps in substituting values correctly.

Answer 1

The Correct Option is B

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:

\[ \boxed{24}. \]