A unit radial vector `hatr` makes angles of `alpha = 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. `(1)/(2)hat(i )+(sqrt(3))/(2)hat(j)`
B. `(sqrt(3))/(2)hat(i )+(1)/(2)hat(j)`
C. `(sqrt(2))/(3)hat(i )+(1)/(sqrt(3))hat(j)`
D. None of these

Answer

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

### Step 1: Understand the relationship between angles and direction cosines
The direction cosines \(L\), \(M\), and \(N\) of a vector are defined as:
- \(L = \cos(\alpha)\)
- \(M = \cos(\beta)\)
- \(N = \cos(\gamma)\)

Given:
- \(\alpha = 30^\circ\)
- \(\beta = 60^\circ\)
- \(\gamma = 90^\circ\)

### Step 2: Calculate the direction cosines
Now we can calculate the direction cosines using the cosine values of the given angles:
- \(L = \cos(30^\circ) = \frac{\sqrt{3}}{2}\)
- \(M = \cos(60^\circ) = \frac{1}{2}\)
- \(N = \cos(90^\circ) = 0\)

### Step 3: Write the unit vector in terms of its components
The unit vector \(\hat{r}\) can be expressed in terms of its components along the x, y, and z axes:
\[ \hat{r} = L \hat{i} + M \hat{j} + N \hat{k} \]

### Step 4: Substitute the values of \(L\), \(M\), and \(N\)
Substituting the calculated values into the equation:
\[ \hat{r} = \left(\frac{\sqrt{3}}{2}\right) \hat{i} + \left(\frac{1}{2}\right) \hat{j} + (0) \hat{k} \]

### Step 5: Simplify the expression
Since the \(N\) component is zero, we can simplify the expression:
\[ \hat{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \]

### Final Result
Thus, the unit radial vector \(\hat{r}\) is:
\[ \hat{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j} \]

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