Question

For given vectors \( \mathbf{a} = -\hat{i} + \hat{j} + 2\hat{k} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \), where \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \), then the value of \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} \) is:

• A. -35
• B. -36
• C. -38
• D. -37

Hint:

For cross products, remember to expand the determinant and carefully calculate the components. The dot product can then be found by multiplying corresponding components.

Answer 1

The Correct Option is A

To find the value of \((\mathbf{a} - \mathbf{b}) \cdot \mathbf{d}\), we need to first calculate the necessary vector products.

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

The cross product of two vectors \(\mathbf{a} = -\hat{i} + \hat{j} + 2\hat{k}\) and \(\mathbf{b} = 2\hat{i} - \hat{j} + \hat{k}\) is computed as follows:

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
\(\mathbf{a}\)	-1	1	2
\(\mathbf{b}\)	2	-1	1

The determinant gives:

\(\mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{vmatrix}\)

\(= \hat{i}(1 \cdot 1 - 2 \cdot (-1)) - \hat{j}(-1 \cdot 1 - 2 \cdot 2) + \hat{k}(-1 \cdot (-1) - 1 \cdot 2)\)

\(= \hat{i}(1 + 2) + \hat{j}(1 + 4) + \hat{k}(1 - 2)\)

\(= 3\hat{i} + 5\hat{j} - \hat{k}\)

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

Using vectors \(\mathbf{c} = 3\hat{i} + 5\hat{j} - \hat{k}\) and \(\mathbf{b} = 2\hat{i} - \hat{j} + \hat{k}\):

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
\(\mathbf{c}\)	3	5	-1
\(\mathbf{b}\)	2	-1	1

The determinant gives:

\(\mathbf{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 5 & -1 \\ 2 & -1 & 1 \end{vmatrix}\)

\(= \hat{i}(5 \cdot 1 - (-1) \cdot (-1)) - \hat{j}(3 \cdot 1 - (-1) \cdot 2) + \hat{k}(3 \cdot (-1) - 5 \cdot 2)\)

\(= \hat{i}(5 - 1) + \hat{j}(3 + 2) + \hat{k}(-3 - 10)\)

\(= 4\hat{i} + 5\hat{j} - 13\hat{k}\)

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

First, find \(\mathbf{a} - \mathbf{b}:\)

\(\mathbf{a} - \mathbf{b} = (-\hat{i} + \hat{j} + 2\hat{k}) - (2\hat{i} - \hat{j} + \hat{k})\)

\(= (-1 - 2)\hat{i} + (1 + 1)\hat{j} + (2 - 1)\hat{k}\)

\(= -3\hat{i} + 2\hat{j} + \hat{k}\)

Now, calculate the dot product with \(\mathbf{d}:\)

\((\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} = (-3\hat{i} + 2\hat{j} + \hat{k}) \cdot (4\hat{i} + 5\hat{j} - 13\hat{k})\)

\(= (-3 \cdot 4) + (2 \cdot 5) + (1 \cdot -13)\)

\(= -12 + 10 - 13\)

\(= -35\)

Thus, the value of \((\mathbf{a} - \mathbf{b}) \cdot \mathbf{d}\) is -35.

The correct answer is -35.