A unit radial vector `vecr` makes agles of `a=30^(@)` relative to the x-axis, `beta=60^(@)` relative to the y-axis, and `gamma=90^(@)` relative to the z-axis. The vector `hatr` can be written as :

Options

A. Statement-I is true, Statement-II is true, Statement-II is correct explanation for statement-I
B. Statement-I is true, Statement-II is true, Statement-II is NOT a correct explanation for statement-I
C. Statement-I is true, Statement-II is false.
D. Statement-I is false and Statement-II is true.

Answer

To find the unit radial vector \(\hat{r}\) given the angles \(\alpha = 30^\circ\), \(\beta = 60^\circ\), and \(\gamma = 90^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand Direction Cosines**: The direction cosines of a vector are defined as the cosines of the angles it makes with the coordinate axes. For a unit vector \(\hat{r}\), these are given by:
\[ \cos(\alpha), \cos(\beta), \cos(\gamma) \]
where \(\alpha\) is the angle with the x-axis, \(\beta\) is the angle with the y-axis, and \(\gamma\) is the angle with the z-axis.

2. **Write the Vector in Terms of Direction Cosines**: The unit vector can be expressed as:
\[ \hat{r} = \cos(\alpha) \hat{i} + \cos(\beta) \hat{j} + \cos(\gamma) \hat{k} \]

3. **Substitute the Given Angles**:
- For \(\alpha = 30^\circ\):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
- For \(\beta = 60^\circ\):
\[ \cos(60^\circ) = \frac{1}{2} \]
- For \(\gamma = 90^\circ\):
\[ \cos(90^\circ) = 0 \]

4. **Plug in the Values**: Now substitute these values into the expression for \(\hat{r}\):
\[ \hat{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} + 0 \hat{k} \]

5. **Simplify the Expression**: The final expression for the unit radial vector \(\hat{r}\) is:
\[ \hat{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \]

### Final Answer:
\[ \hat{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \]