A vector is inclined at `(pi)/(4)` rad with x-axis `(pi)/(3)` rad with y-axis then find angle of vector with z-axis

Options

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

Answer

To find the angle of a vector with the z-axis given its angles with the x-axis and y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Angles**:
- The vector makes an angle of \(\frac{\pi}{4}\) radians with the x-axis.
- The vector makes an angle of \(\frac{\pi}{3}\) radians with the y-axis.
- We need to find the angle \(\theta\) that the vector makes with the z-axis.

2. **Use the Cosine Law for Angles**:
- For a vector \( \vec{A} \) in three-dimensional space, we can express it in terms of its angles with the axes:
\[ \vec{A} = A \cos\left(\frac{\pi}{4}\right) \hat{i} + A \cos\left(\frac{\pi}{3}\right) \hat{j} + A \cos(\theta) \hat{k} \]
- Here, \( A \) is the magnitude of the vector, and \( \hat{i}, \hat{j}, \hat{k} \) are the unit vectors along the x, y, and z axes, respectively.

3. **Magnitude of the Vector**:
- The magnitude of the vector can also be expressed using the Pythagorean theorem:
\[ A = \sqrt{(A \cos\left(\frac{\pi}{4}\right))^2 + (A \cos\left(\frac{\pi}{3}\right))^2 + (A \cos(\theta))^2} \]
- Simplifying this, we get:
\[ A = A \sqrt{\cos^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{3}\right) + \cos^2(\theta)} \]

4. **Calculate the Cosines**:
- Calculate \( \cos\left(\frac{\pi}{4}\right) \) and \( \cos\left(\frac{\pi}{3}\right) \):
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

5. **Substitute the Values**:
- Substitute the values into the equation:
\[ A = A \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2(\theta)} \]
- This simplifies to:
\[ A = A \sqrt{\frac{1}{2} + \frac{1}{4} + \cos^2(\theta)} \]

6. **Cancel \( A \) and Rearrange**:
- Cancel \( A \) (assuming \( A \neq 0 \)):
\[ 1 = \sqrt{\frac{1}{2} + \frac{1}{4} + \cos^2(\theta)} \]
- Square both sides:
\[ 1 = \frac{1}{2} + \frac{1}{4} + \cos^2(\theta) \]

7. **Combine Terms**:
- Combine the fractions:
\[ 1 = \frac{2}{4} + \frac{1}{4} + \cos^2(\theta) \implies 1 = \frac{3}{4} + \cos^2(\theta) \]
- Rearranging gives:
\[ \cos^2(\theta) = 1 - \frac{3}{4} = \frac{1}{4} \]

8. **Find \( \cos(\theta) \)**:
- Taking the square root:
\[ \cos(\theta) = \pm \frac{1}{2} \]

9. **Determine Possible Angles**:
- The angles corresponding to \( \cos(\theta) = \frac{1}{2} \) are:
\[ \theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{5\pi}{3} \]
- The angle corresponding to \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = \frac{2\pi}{3} \]

10. **Final Answer**:
- The angle of the vector with the z-axis is:
\[ \theta = \frac{2\pi}{3} \text{ radians} \]