If `vec(P) = hat(i) + hat(j) + hat(k)`, its direction cosines are

Options

A. 1,1,1
B. `1//sqrt(3), 1//sqrt(3), 1//sqrt(3)`
C. `sqrt(3), sqrt(3), sqrt(3)`
D. 0,0,0

Answer

To find the direction cosines of the vector \(\vec{P} = \hat{i} + \hat{j} + \hat{k}\), we can follow these steps:

### Step 1: Identify the components of the vector
The vector \(\vec{P}\) can be expressed in terms of its components along the x, y, and z axes:
- The coefficient of \(\hat{i}\) (x-component) is \(a = 1\).
- The coefficient of \(\hat{j}\) (y-component) is \(b = 1\).
- The coefficient of \(\hat{k}\) (z-component) is \(c = 1\).

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \(\vec{P}\) is given by:
\[ |\vec{P}| = \sqrt{a^2 + b^2 + c^2} = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

### Step 3: Calculate the direction cosines
The direction cosines \(\alpha\), \(\beta\), and \(\gamma\) are defined as:
\[ \alpha = \frac{a}{|\vec{P}|}, \quad \beta = \frac{b}{|\vec{P}|}, \quad \gamma = \frac{c}{|\vec{P}|} \]
Substituting the values:
- For \(\alpha\):
\[ \alpha = \frac{1}{\sqrt{3}} \]
- For \(\beta\):
\[ \beta = \frac{1}{\sqrt{3}} \]
- For \(\gamma\):
\[ \gamma = \frac{1}{\sqrt{3}} \]

### Step 4: Write the final result
The direction cosines of the vector \(\vec{P}\) are:
\[ \alpha = \frac{1}{\sqrt{3}}, \quad \beta = \frac{1}{\sqrt{3}}, \quad \gamma = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the direction cosines are \(\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\).

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