What is the sum of the squares of direction cosines of X-axis?

Options

A. 0
B. `1/3`
C. 1
D. 3

Answer

To find the sum of the squares of the direction cosines of the X-axis, we can follow these steps:

### Step 1: Identify the Direction Ratios of the X-axis
The direction ratios for the X-axis are given as:
- \( x = 1 \)
- \( y = 0 \)
- \( z = 0 \)

### Step 2: Calculate the Magnitude of the Direction Ratios
The magnitude (or length) of the direction ratios can be calculated using the formula:
\[ \text{Magnitude} = \sqrt{x^2 + y^2 + z^2} \]
Substituting the values:
\[ \text{Magnitude} = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1 \]

### Step 3: Calculate the Direction Cosines
The direction cosines are calculated using the formula:
\[ \cos \alpha = \frac{x}{\text{Magnitude}}, \quad \cos \beta = \frac{y}{\text{Magnitude}}, \quad \cos \gamma = \frac{z}{\text{Magnitude}} \]
Substituting the values:
\[ \cos \alpha = \frac{1}{1} = 1, \quad \cos \beta = \frac{0}{1} = 0, \quad \cos \gamma = \frac{0}{1} = 0 \]

### Step 4: Sum of the Squares of the Direction Cosines
Now, we need to find the sum of the squares of the direction cosines:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \]
Substituting the values:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1 \]

### Final Answer
The sum of the squares of the direction cosines of the X-axis is:
\[ \boxed{1} \]

---