If a vector `vecP` makes an angle `alpha, beta, gamma " with " x,y,z` axis respectively then it can be represented as `vecP=P[cos alphahati+cos beta hatj+ cos gamma hatk]`. Choose the correct option(s) :-\

Options

A. `cos^(2) alpha + cos^(2) beta +cos^(2)gamma=1`
B. `vecP. vecP=P^(2)`
C. `vecP.(hati-hatk)=P(cos alpha-cos gamma)`
D. `vecP. hati= cos alpha`

Answer

To solve the question regarding the vector \(\vec{P}\) and its representation in terms of angles with the axes, we will analyze the provided options step by step.

### Step-by-Step Solution

1. **Understanding the Vector Representation**:
The vector \(\vec{P}\) is represented as:
\[ \vec{P} = P (\cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}) \]
where \(\alpha\), \(\beta\), and \(\gamma\) are the angles that the vector makes with the x, y, and z axes, respectively.

2. **Direction Cosines**:
The terms \(\cos \alpha\), \(\cos \beta\), and \(\cos \gamma\) are known as the direction cosines of the vector \(\vec{P}\). They have a fundamental property:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
This is a well-known relation for direction cosines, confirming that option 1 is correct.

3. **Dot Product of the Vector**:
The dot product of a vector with itself is given by:
\[ \vec{P} \cdot \vec{P} = P^2 \]
Therefore, we can conclude that:
\[ \vec{P} \cdot \vec{P} = P^2 \]
This confirms that option 2 is also correct.

4. **Analyzing Option 3**:
We need to verify the expression:
\[ \vec{P} (\hat{i} - \hat{k}) = P (\cos \alpha - \cos \gamma) \]
Expanding the left-hand side:
\[ \vec{P} (\hat{i} - \hat{k}) = P (\cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k})(\hat{i} - \hat{k}) \]
This simplifies to:
\[ P (\cos \alpha \hat{i} - \cos \gamma \hat{k}) = P (\cos \alpha - \cos \gamma) \]
Thus, option 3 is also correct.

5. **Evaluating Option 4**:
The statement in option 4 is:
\[ \vec{P} \cdot \hat{i} = \cos \alpha \]
However, the correct expression is:
\[ \vec{P} \cdot \hat{i} = P \cos \alpha \]
Therefore, option 4 is incorrect.

### Conclusion
The correct options are:
- Option 1: \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)
- Option 2: \(\vec{P} \cdot \vec{P} = P^2\)
- Option 3: \(\vec{P} (\hat{i} - \hat{k}) = P (\cos \alpha - \cos \gamma)\)