If unit vector a makes angles `pi/3` with i, `pi//4` with j and `theta in (0,pi)` with k, then a value of `theta` is

Options

A. `(2pi)/3`
B. `(5pi)/6`
C. `(5pi)/12`
D. `pi/4`

Answer

To solve the problem, we need to find the value of the angle \( \theta \) that the unit vector \( \mathbf{a} \) makes with the k-axis, given that it makes angles \( \frac{\pi}{3} \) with the i-axis and \( \frac{\pi}{4} \) with the j-axis.

### Step-by-step Solution:

1. **Understanding the Unit Vector**:
A unit vector \( \mathbf{a} \) can be expressed in terms of its components along the i, j, and k axes:
\[ \mathbf{a} = A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k} \]
where \( A_1, A_2, A_3 \) are the components along the i, j, and k axes, respectively.

2. **Using the Cosine of Angles**:
The cosine of the angle that a vector makes with an axis is given by the ratio of the component along that axis to the magnitude of the vector. Since \( \mathbf{a} \) is a unit vector, its magnitude is 1.

- For the angle \( \frac{\pi}{3} \) with the i-axis:
\[ \cos\left(\frac{\pi}{3}\right) = A_1 \]
Thus, \( A_1 = \frac{1}{2} \).

- For the angle \( \frac{\pi}{4} \) with the j-axis:
\[ \cos\left(\frac{\pi}{4}\right) = A_2 \]
Thus, \( A_2 = \frac{1}{\sqrt{2}} \).

3. **Finding the Component Along the k-axis**:
The angle \( \theta \) with the k-axis gives us:
\[ \cos(\theta) = A_3 \]

4. **Using the Unit Vector Condition**:
Since \( \mathbf{a} \) is a unit vector, we have:
\[ A_1^2 + A_2^2 + A_3^2 = 1 \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + A_3^2 = 1 \]
Simplifying this gives:
\[ \frac{1}{4} + \frac{1}{2} + A_3^2 = 1 \]
\[ \frac{1}{4} + \frac{2}{4} + A_3^2 = 1 \]
\[ \frac{3}{4} + A_3^2 = 1 \]
\[ A_3^2 = 1 - \frac{3}{4} = \frac{1}{4} \]
Therefore, \( A_3 = \pm \frac{1}{2} \).

5. **Finding the Value of \( \theta \)**:
Since \( A_3 = \cos(\theta) \), we have:
\[ \cos(\theta) = \frac{1}{2} \quad \text{or} \quad \cos(\theta) = -\frac{1}{2} \]
The values of \( \theta \) that satisfy \( \cos(\theta) = \frac{1}{2} \) are:
\[ \theta = \frac{\pi}{3} \quad \text{(in the range } (0, \pi)\text{)} \]
The value of \( \theta \) that satisfies \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = \frac{2\pi}{3} \quad \text{(also in the range } (0, \pi)\text{)} \]

### Conclusion:
Thus, the possible values of \( \theta \) are \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \).