In a $p$-dimensional space, a hyperplane is defined mathematically by the equation $\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_p X_p = 0$. Any point $X = (X_1, X_2, \dots, X_p)$ that satisfies this equation lies on the hyperplane. For example, in two dimensions ($p=2$), this equation reduces to $\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0$, which represents a one-dimensional hyperplane, commonly known as a line.