Yes, the residuals might be both positive and negative. The linear regression typically minimizes the square of them.

In case of two-dimensional input, we obtain a regression plane and the residuals are calculated in the same way.

Yes, the residuals might be both positive and negative. The linear regression typically minimizes the square of them.

In case of two-dimensional input, we obtain a regression plane and the residuals are calculated in the same way.

EDIT: The regression plane is defined as $$ z_i =\beta_0+\beta_1x_{i} +\beta_2y_{i}+\epsilon_i $$ and the residual is for given parameters $\beta_0,\beta_1,\beta_2$ and given data record $(z_i,y_i,x_i)$ calculated as $$ \epsilon_i=z_i -(\beta_0+\beta_1x_{i} +\beta_2y_{i}) $$ Similarly also with higher dimensions.