Question

If a and b be two perpendicular unit vectors such that \(\mathbf{x} = \mathbf{b} - (\mathbf{a} \times \mathbf{x})\), then \(|\mathbf{x}|\) is equal to

• A. 1
• B. \(\sqrt{2}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. \(\sqrt{3}\)

Hint:

Using vector triple product identities and dot product properties simplifies such vector equations.

Answer 1

The Correct Option is C

To solve the equation given, we need to understand the vector operations involved:

1. We are given \(\mathbf{x} = \mathbf{b} - (\mathbf{a} \times \mathbf{x})\).
2. This involves the cross product \(\mathbf{a} \times \mathbf{x}\) and we know that \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors and perpendicular to each other. Therefore, \(\mathbf{a} \cdot \mathbf{b} = 0\).
3. Rewrite the equation as: \(\mathbf{x} + (\mathbf{a} \times \mathbf{x}) = \mathbf{b}\).
4. Since \(\mathbf{a} \cdot \mathbf{b} = 0\), \(\mathbf{b}\) actually lies in the plane perpendicular to \(\mathbf{a}\), represented by the vector \(\mathbf{x} + \mathbf{a} \times \mathbf{x}\).
5. The magnitude of the expression \(|\mathbf{x} + \mathbf{a} \times \mathbf{x}|\) can be calculated using Pythagoras' theorem for vectors, assuming perpendicular components. Let:
   • Let \(|\mathbf{x}| = r\),
   • Then \(|(\mathbf{a} \times \mathbf{x})| = |\mathbf{a}||\mathbf{x}|\sin(\theta) = r\sin(\theta)\) (Since \(|\mathbf{a}|\) is 1 and \(\theta\) is 90 degrees when working with standard axis vectors).
6. Therefore: \(r \sqrt{1 + (\sin(\theta))^2} = |\mathbf{b}| = 1 \quad (\text{since } |\mathbf{b}| = 1)\).
7. Solving: \(r \sqrt{2} = 1\; \Rightarrow \; r = \frac{1}{\sqrt{2}}\).
8. Hence, the magnitude of the vector \(|\mathbf{x}|\) is \(\frac{1}{\sqrt{2}}\).

The correct answer is therefore \(\frac{1}{\sqrt{2}}\)