# Obtaining the $j$th component of the OLS - an explanation [duplicate]

In a linear regression setting, I've seen that the $$j$$th component of the ordinary least square estimator $$\hat{\beta}_j$$ can be obtained as follows: $$\hat{\beta}_j = Y^T Z^{(j)} / (X^{(j)})^T Z^{(j)},$$ where $$Y$$ is the response vector, $$X^{(j)}$$ is the $$j$$th column of the design matrix, and $$Z^{(j)}$$ is the residual vector when regressing $$X^{(j)}$$ against the remaining columns of the design matrix $$X^{(-j)}$$.

Could you help me understand why this works? A geometric explanation would be really helpful!