Question

A vector \( \vec{A} = i + xj + 3k \) is rotated through an angle and is also doubled in magnitude resulting in \( \vec{B} = 4i + (4x - 2)j + 2k \). An acceptable value of x is

• A. 1
• B. 2
• C. 3
• D. \( \frac{4}{3} \)

Hint:

When solving problems involving vector magnitudes and rotations, always check both the magnitude and direction of the vector components.

Answer 1

The Correct Option is B

Step 1: Understanding vector magnitude and components.
The magnitude of vector \( \vec{A} \) is given by \( \sqrt{1^2 + x^2 + 3^2} \). After doubling the magnitude and rotating, the magnitude of \( \vec{B} \) should be \( 2 \times \text{magnitude of } \vec{A} \). Compare the components of \( \vec{A} \) and \( \vec{B} \) to find the value of \( x \).

Step 2: Analyzing the options.
By comparing the components of \( \vec{A} \) and \( \vec{B} \), we can find that \( x = 2 \).

Step 3: Conclusion.
The correct answer is (B) as \( x = 2 \) satisfies the given condition.