Question

If \(\mathbf{a} = -\hat{\mathbf{i}} + 2\hat{\mathbf{j}} - \hat{\mathbf{k}}\), \(\mathbf{b} = \hat{\mathbf{i}} + \hat{\mathbf{j}} - 3\hat{\mathbf{k}}\) and \(\mathbf{c} = -4\hat{\mathbf{i}} - \hat{\mathbf{k}}\), then \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\) is

• A. \(5\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 15\hat{\mathbf{k}}\)
• B. 0
• C. \(12\hat{\mathbf{j}} + 4\hat{\mathbf{k}}\)
• D. \(-3\hat{\mathbf{i}} + 6\hat{\mathbf{j}} - 3\hat{\mathbf{k}}\)

Hint:

Vector triple product: \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\).

Answer 1

The Correct Option is A

To solve the given vector expression \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\), we will perform the following steps:

1. Calculate the cross product \(\mathbf{b} \times \mathbf{c}\).
2. Use the result from step 1 to find \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\).
3. Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
4. Multiply the result from step 3 with vector \(\mathbf{c}\).
5. Add the results from steps 2 and 4.

Step-by-Step Solution

Step 1: Calculate \(\mathbf{b} \times \mathbf{c}\)

The vectors are given as:

\[ \mathbf{b} = \hat{\mathbf{i}} + \hat{\mathbf{j}} - 3\hat{\mathbf{k}}, \quad \mathbf{c} = -4\hat{\mathbf{i}} - \hat{\mathbf{k}} \]

The cross product is given by:

\(\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & 1 & -3 \\ -4 & 0 & -1 \end{vmatrix}\)

Expanding this determinant, we get:

\[ \mathbf{b} \times \mathbf{c} = \hat{\mathbf{i}}(1 \cdot (-1) - (0 \cdot -3)) - \hat{\mathbf{j}}(1 \cdot (-1) - (-3 \cdot -4)) + \hat{\mathbf{k}}(1 \cdot 0 - 1 \cdot -4) \]

Simplifying, we find:

\[ = \hat{\mathbf{i}}(-1) - \hat{\mathbf{j}}(-1 - 12) + \hat{\mathbf{k}}(4) \\ = -\hat{\mathbf{i}} + 13\hat{\mathbf{j}} + 4\hat{\mathbf{k}} \]

Step 2: Calculate \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\)

Now, using the result from Step 1 and vector \(\mathbf{a} = -\hat{\mathbf{i}} + 2\hat{\mathbf{j}} - \hat{\mathbf{k}}\), calculate:

\(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -1 & 2 & -1 \\ -1 & 13 & 4 \end{vmatrix}\)

Expanding the determinant gives:

\[ = \hat{\mathbf{i}}(2 \cdot 4 - (-1 \cdot 13)) - \hat{\mathbf{j}}((-1) \cdot 4 - (-1 \cdot -1)) + \hat{\mathbf{k}}((-1) \cdot 13 - 2 \cdot -1) \]

Simplifying, we get:

\[ = \hat{\mathbf{i}}(8 + 13) - \hat{\mathbf{j}}(-4 - 1) + \hat{\mathbf{k}}(-13 + 2) \\ = 21\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 11\hat{\mathbf{k}} \]

Step 3: Calculate \(\mathbf{a} \cdot \mathbf{b}\)

Find the dot product of \(\mathbf{a}\) and \(\mathbf{b}\):

\[ \mathbf{a} \cdot \mathbf{b} = (-1 \cdot 1) + (2 \cdot 1) + (-1 \cdot -3) \\ = -1 + 2 + 3 = 4 \]

Step 4: Multiply the result from Step 3 with vector \(\mathbf{c}\)

Multiply the scalar with vector \(\mathbf{c}\):

\[ 4 \mathbf{c} = 4(-4\hat{\mathbf{i}} - \hat{\mathbf{k}}) \\ = -16\hat{\mathbf{i}} - 4\hat{\mathbf{k}} \]

Step 5: Add the results from Steps 2 and 4

Add the vector results from Step 2 and Step 4:

\[ (21\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 11\hat{\mathbf{k}}) + (-16\hat{\mathbf{i}} - 4\hat{\mathbf{k}}) \\ = (21 - 16)\hat{\mathbf{i}} + 5\hat{\mathbf{j}} + (-11 - 4)\hat{\mathbf{k}} \\ = 5\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 15\hat{\mathbf{k}} \]

Conclusion

The result of the expression \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}\) is \(\mathbf{5\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 15\hat{\mathbf{k}}}\), which corresponds to the correct option.