Question

If \( \bar{a} = \bar{i} - 2\bar{j} - 2\bar{k} \) and \( \bar{b} = 2\bar{i} + \bar{j} + 2\bar{k} \) are two vectors then \( (\bar{a} + 2\bar{b}) \times (3\bar{a} - \bar{b}) = \)

• A. \( 2\bar{i} + 6\bar{j} - 5\bar{k} \)
• B. \( 6\bar{i} - 2\bar{j} + 3\bar{k} \)
• C. \( 14\bar{i} + 7\bar{j} - 5\bar{k} \)
• D. \( 14\bar{i} + 42\bar{j} - 35\bar{k} \)

Hint:

Simplifying the vector algebraic expression before substituting components minimizes the arithmetic and reduces the chance of sign errors in the determinant calculation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We need to compute the cross product of two composite vectors. Instead of substituting the components immediately, it is more efficient to use the algebraic properties of the cross product (distributivity and anticommutativity) to simplify the expression first.
Step 2: Key Formula or Approach:

Properties of cross product: 1. \( \bar{u} \times (\bar{v} + \bar{w}) = \bar{u} \times \bar{v} + \bar{u} \times \bar{w} \) 2. \( \bar{u} \times \bar{u} = \vec{0} \) 3. \( \bar{v} \times \bar{u} = -(\bar{u} \times \bar{v}) \)
Step 3: Detailed Explanation:

Simplify the expression: \[ (\bar{a} + 2\bar{b}) \times (3\bar{a} - \bar{b}) = \bar{a} \times (3\bar{a}) - \bar{a} \times \bar{b} + 2\bar{b} \times (3\bar{a}) - 2\bar{b} \times \bar{b} \] \[ = 3(\bar{a} \times \bar{a}) - (\bar{a} \times \bar{b}) + 6(\bar{b} \times \bar{a}) - 2(\bar{b} \times \bar{b}) \] Since \( \bar{a} \times \bar{a} = 0 \) and \( \bar{b} \times \bar{b} = 0 \): \[ = 0 - (\bar{a} \times \bar{b}) - 6(\bar{a} \times \bar{b}) - 0 \] (Using \( \bar{b} \times \bar{a} = -(\bar{a} \times \bar{b}) \)) \[ = -7(\bar{a} \times \bar{b}) \] Now, calculate \( \bar{a} \times \bar{b} \) using the determinant method: \[ \bar{a} \times \bar{b} = \begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ 1 & -2 & -2 \\ 2 & 1 & 2 \end{vmatrix} \] \[ = \bar{i}((-2)(2) - (-2)(1)) - \bar{j}((1)(2) - (-2)(2)) + \bar{k}((1)(1) - (-2)(2)) \] \[ = \bar{i}(-4 + 2) - \bar{j}(2 + 4) + \bar{k}(1 + 4) \] \[ = -2\bar{i} - 6\bar{j} + 5\bar{k} \] Finally, multiply by -7: \[ -7(\bar{a} \times \bar{b}) = -7(-2\bar{i} - 6\bar{j} + 5\bar{k}) \] \[ = 14\bar{i} + 42\bar{j} - 35\bar{k} \]
Step 4: Final Answer:

The result is \( 14\bar{i} + 42\bar{j} - 35\bar{k} \).