The vector ` hat i+x hat j+3 hat k` is rotated through an angle `theta` and doubled in magnitude, then it becomes `4 hat i+(4x-2)dot hat j+2 hat k` . Then value of `x` are `-2/3` (b) `1/3` (c) `2/3` (d) 2

Options

A. `{-(2)/(3),2}`
B. `((1)/(3),2)`
C. `{(2)/(3),0}`
D. `{2,7}`

Explanation

To solve the problem, we start with the given vectors and the conditions provided.

1. **Define the vectors**:
Let vector \( \mathbf{A} = \hat{i} + x \hat{j} + 3 \hat{k} \) and vector \( \mathbf{C} = 4 \hat{i} + (4x - 2) \hat{j} + 2 \hat{k} \).

2. **Magnitude of the vectors**:
The magnitude of vector \( \mathbf{A} \) is given by:
\[
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