What is the length of the vector (1, 1) ?

Options

A. 0
B. 1
C. `sqrt(2)`
D. `1/2`

Explanation

To find the length of the vector \( \mathbf{v} = (1, 1) \), we can use the formula for the length (or magnitude) of a vector in two dimensions. The formula is given by:

\[ \text{Length} = \sqrt{x^2 + y^2} \]

where \( x \) and \( y \) are the components of the vector.

### Step-by-step Solution:

1. **Identify the components of the vector**:
The vector given is \( \mathbf{v} = (1, 1) \). Here, \( x = 1 \) and \( y = 1 \).

2. **Substitute the components into the formula**:
We substitute \( x \) and \( y \) into the length formula:
\[ \text{Length} = \sqrt{1^2 + 1^2} \]

3. **Calculate \( x^2 \) and \( y^2 \)**:
\[ 1^2 = 1 \quad \text{and} \quad 1^2 = 1 \]

4. **Add the squares**:
\[ 1 + 1 = 2 \]

5. **Take the square root**:
\[ \text{Length} = \sqrt{2} \]

Thus, the length of the vector \( (1, 1) \) is \( \sqrt{2} \).

### Final Answer:
The length of the vector \( (1, 1) \) is \( \sqrt{2} \).