The component of a vector `r` along X-axis will have maximum value if

Options

A. `vec(r )` is along +ve x - axis
B. `vec(r )` is along +ve y - axis
C. `vec(r )` is along -ve y - axis
D. `vec(r )` makes an angle of `45^(@)` with the x-axis

Answer

To solve the problem of finding the condition under which the component of a vector \( \mathbf{r} \) along the X-axis has a maximum value, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Vector Components**:
- A vector \( \mathbf{r} \) can be represented in a coordinate system where it makes an angle \( \theta \) with the X-axis. The components of the vector can be expressed in terms of its magnitude \( r \) and the angle \( \theta \).

2. **Expression for X-axis Component**:
- The component of the vector \( \mathbf{r} \) along the X-axis is given by the formula:
\[ r_x = r \cos \theta \]
- Here, \( r_x \) is the component of the vector along the X-axis, \( r \) is the magnitude of the vector, and \( \theta \) is the angle between the vector and the X-axis.

3. **Analyzing the Cosine Function**:
- The cosine function \( \cos \theta \) varies between -1 and 1. It reaches its maximum value of 1 when \( \theta = 0^\circ \).
- Therefore, to maximize \( r_x \), we need to maximize \( \cos \theta \).

4. **Condition for Maximum Component**:
- The maximum value of \( r_x \) occurs when:
\[ \cos \theta = 1 \quad \text{at} \quad \theta = 0^\circ \]
- This means that the vector \( \mathbf{r} \) must be aligned along the positive X-axis.

5. **Conclusion**:
- Thus, the component of the vector \( \mathbf{r} \) along the X-axis will have its maximum value when the vector is directed along the positive X-axis.

### Final Answer:
The component of a vector \( \mathbf{r} \) along the X-axis will have a maximum value if the vector is along the positive X-axis (i.e., \( \theta = 0^\circ \)).

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