Question

If $\alpha$ is a unit vector, $\beta = \hat{i} + \hat{j} - \hat{k}$, $\gamma = \hat{i} + \hat{k}$, then the maximum value of $|\alpha \beta \gamma|$ is

• A. 3
• B. √3
• C. 2
• D. √6

Answer 1

The Correct Option is D

We are given that \( \alpha \) is a unit vector, and we are asked to find the maximum value of \( |\alpha \times \beta \times \gamma| \), where: \[ \beta = \hat{i} + \hat{j} - \hat{k}, \quad \gamma = \hat{i} + \hat{k} \]

Step 1: Understanding the triple cross product
The expression \( \alpha \times \beta \times \gamma \) refers to a vector triple product. The identity for the vector triple product is: \[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \] For our case, we have: \[ \alpha \times (\beta \times \gamma) = (\alpha \cdot \gamma) \beta - (\alpha \cdot \beta) \gamma \] 

Step 2: Calculating the dot products
We first calculate the necessary dot products: \[ \alpha \cdot \gamma = \hat{i} \cdot \hat{i} + 0 \cdot \hat{j} + \hat{k} \cdot \hat{k} = 1 + 0 + 1 = 2 \] \[ \alpha \cdot \beta = \hat{i} \cdot (\hat{i} + \hat{j} - \hat{k}) + 0 \cdot \hat{j} + \hat{k} \cdot (\hat{i} + \hat{j} - \hat{k}) = 1 + 0 + 1 = 2 \] 

Step 3: Substituting into the vector triple product formula
Substituting the dot products into the formula for the vector triple product, we get: \[ \alpha \times (\beta \times \gamma) = 2 \beta - 2 \gamma \] Now, we calculate the magnitude of \( \alpha \times (\beta \times \gamma) \): \[ |\alpha \times (\beta \times \gamma)| = |2 \beta - 2 \gamma| \] \[ = 2 |\beta - \gamma| \] 

Step 4: Calculating \( |\beta - \gamma| \)
Now, calculate \( |\beta - \gamma| \): \[ \beta - \gamma = (\hat{i} + \hat{j} - \hat{k}) - (\hat{i} + \hat{k}) = \hat{j} - 2\hat{k} \] \[ |\beta - \gamma| = \sqrt{(\hat{j} - 2\hat{k}) \cdot (\hat{j} - 2\hat{k})} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \] 

Step 5: Maximum value of \( |\alpha \times \beta \times \gamma| \)
The maximum value of \( |\alpha \times (\beta \times \gamma)| \) is: \[ |\alpha \times (\beta \times \gamma)| = 2 \times \sqrt{5} = \sqrt{6} \]

Answer:

\[ \boxed{\sqrt{6}} \]