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How to produce the mathematical hat symbol or and a wide hat symbol ? \hat and \widehat

The \hat and \widehat commands are used in LaTeX to produce the "hat" accent over a symbol, indicating a variable quantity or an estimate of a parameter.

The \hat command produces a small hat, while the \widehat command produces a wider hat that spans the entire width of the symbol.

In mathematics and physics, the hat symbol is often used to represent unit vectors, which have a magnitude of 1 and are used to describe the direction of a physical quantity. For example, the unit vector in the x direction is denoted as $\hat{i}$, while the unit vector in the y direction is denoted as $\hat{j}$:

```
$$
\hat{i} = (1, 0, 0) \quad \widehat{\mathbf{r}} = (x, y, z)/\sqrt{x^2 + y^2 + z^2}
$$
```

$$ \hat{i} = (1, 0, 0) \quad \widehat{\mathbf{r}} = (x, y, z)/\sqrt{x^2 + y^2 + z^2} $$

In statistics, the hat symbol is often used to denote an estimator of a parameter. For example, the sample mean is denoted as $\hat{\mu}$, which is an estimate of the population mean:

```
$$
\hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i
$$
```

$$ \hat{\mu} = \frac{1}{n}\sum_{i=1}^n x_i $$

In geometry, the hat symbol is used to denote the altitude of a triangle, which is a line segment drawn from a vertex of the triangle perpendicular to the opposite side:

```
$$
h_a = \widehat{BC}
$$
```

$$ h_a = \widehat{BC} $$

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