The modulus of the sum of three mutually perpendicular unit vectors is

Options

A. `sqrt(3)`
B. 3
C. 0
D. none

Answer

To find the modulus of the sum of three mutually perpendicular unit vectors, we can follow these steps:

### Step 1: Identify the unit vectors
The three mutually perpendicular unit vectors can be represented as:
- \(\mathbf{i} = \hat{i}\)
- \(\mathbf{j} = \hat{j}\)
- \(\mathbf{k} = \hat{k}\)

### Step 2: Write the sum of the vectors
We need to find the sum of these vectors:
\[ \mathbf{S} = \hat{i} + \hat{j} + \hat{k} \]

### Step 3: Calculate the modulus squared of the sum
To find the modulus, we first calculate the square of the modulus:
\[ |\mathbf{S}|^2 = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) \]

### Step 4: Expand the dot product
Using the distributive property of the dot product:
\[ |\mathbf{S}|^2 = \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} + 2(\hat{i} \cdot \hat{j} + \hat{j} \cdot \hat{k} + \hat{k} \cdot \hat{i}) \]

### Step 5: Substitute the values
Since \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) are unit vectors:
- \(\hat{i} \cdot \hat{i} = 1\)
- \(\hat{j} \cdot \hat{j} = 1\)
- \(\hat{k} \cdot \hat{k} = 1\)

And since they are mutually perpendicular:
- \(\hat{i} \cdot \hat{j} = 0\)
- \(\hat{j} \cdot \hat{k} = 0\)
- \(\hat{k} \cdot \hat{i} = 0\)

Thus, we have:
\[ |\mathbf{S}|^2 = 1 + 1 + 1 + 2(0 + 0 + 0) = 3 \]

### Step 6: Take the square root to find the modulus
Now, we take the square root of both sides:
\[ |\mathbf{S}| = \sqrt{3} \]

### Conclusion
The modulus of the sum of the three mutually perpendicular unit vectors is \(\sqrt{3}\).

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