If` veca,vecb,vecc` are three mutually perpendicular vectors, then the vector which is equally inclined to these vectors is (A) `veca+vecb+vecc` (B) `veca/|veca|+vecb/|vecb|+vec/|vecc|` (C) `veca/|veca|^2+vecb/|vecb|^2+vecc/|vecc|^2` (D) `|veca|veca-|vecb|vecb+|vecc|vecc`

Options

A. `veca+vecb+vecc`
B. `veca/|veca|+vecb/|vecb|+vecc/|vecc|`
C. `veca/|veca|^(2)+vecb/|vecb|^(2)+vecc/|vecc|^(2)`
D. `|veca|veca-|vecb|vecb+|vecc|vecc`

Explanation

To find the vector that is equally inclined to three mutually perpendicular vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can follow these steps:

### Step 1: Understanding the Problem
We have three mutually perpendicular vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). We need to find a vector \(\vec{\alpha}\) that makes equal angles with all three vectors.

### Step 2: Expressing the Vector
To express the vector \(\vec{\alpha}\) that is equally inclined to \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can consider the unit vectors in the direction of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):
\[ \vec{\alpha} = \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} + \frac{\vec{c}}{|\vec{c}|} \]

### Step 3: Justifying Equal Angles
Since \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are mutually perpendicular, the angles between \(\vec{\alpha}\) and each of these vectors will be equal. This is because the dot product will yield the same cosine value for each angle, leading to:
\[ \cos(\theta) = \frac{\vec{\alpha} \cdot \vec{a}}{|\vec{\alpha}| |\vec{a}|} \]
\[ \cos(\phi) = \frac{\vec{\alpha} \cdot \vec{b}}{|\vec{\alpha}| |\vec{b}|} \]
\[ \cos(\psi) = \frac{\vec{\alpha} \cdot \vec{c}}{|\vec{\alpha}| |\vec{c}|} \]
Since \(\vec{\alpha}\) is constructed from the unit vectors of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), it will maintain equal angles with all three.

### Step 4: Conclusion
Thus, the vector that is equally inclined to the three mutually perpendicular vectors is:
\[ \vec{\alpha} = \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} + \frac{\vec{c}}{|\vec{c}|} \]
This corresponds to option (B).

### Final Answer
The correct answer is (B) \(\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} + \frac{\vec{c}}{|\vec{c}|}\).

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