The wavefronts of a light wave travellilng in vacuum are given by `x+y+z=c`. The angle made by the direction of propagation of light with the X-axis is

Options

A. `0^@`
B. `45^@`
C. `90^@`
D. `cos^(-1) (1//sqrt3).`

Answer

To find the angle made by the direction of propagation of light with the X-axis given the wavefront equation \(x + y + z = c\), we can follow these steps:

### Step 1: Understand the Equation of the Wavefront
The equation \(x + y + z = c\) represents a plane in three-dimensional space. The normal vector to this plane can be derived from the coefficients of \(x\), \(y\), and \(z\).

**Hint:** The normal vector to a plane \(Ax + By + Cz = D\) is given by the vector \((A, B, C)\).

### Step 2: Identify the Normal Vector
From the equation \(x + y + z = c\), the coefficients are:
- \(A = 1\)
- \(B = 1\)
- \(C = 1\)

Thus, the normal vector \( \vec{n} \) to the plane is:
\[ \vec{n} = (1, 1, 1) \]

**Hint:** The direction of propagation of the wave is along the normal vector of the wavefront.

### Step 3: Calculate the Direction Cosines
The direction cosines of the normal vector are given by:
\[ \cos \alpha = \frac{1}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{1}{\sqrt{3}} \]
\[ \cos \beta = \frac{1}{\sqrt{3}}, \quad \cos \gamma = \frac{1}{\sqrt{3}} \]

**Hint:** The direction cosines are the cosines of the angles that the vector makes with the coordinate axes.

### Step 4: Use the Property of Direction Cosines
The sum of the squares of the direction cosines must equal 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Substituting the values:
\[ \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 = 1 \]
This simplifies to:
\[ 3 \cdot \frac{1}{3} = 1 \]

**Hint:** This property confirms that the values calculated for direction cosines are correct.

### Step 5: Calculate the Angle with the X-axis
To find the angle \( \theta \) made with the X-axis, we use:
\[ \cos \theta = \frac{1}{\sqrt{3}} \]
Thus, the angle \( \theta \) is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

**Hint:** Use a calculator or trigonometric tables to find the angle if needed.

### Final Answer
The angle made by the direction of propagation of light with the X-axis is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

**Hint:** Check the options provided in the question to find the correct match for the calculated angle.