Suppose a line goes through the points A = ( x 1 , y 1 ) and B = ( x 2 , y 2 ). One can easily check (as I did today while doodling) that b = − x 1 y 2 − x 2 y 1 x 2 − x 1 where b is the y-coordinate of the y-intercept. This can be written more suggestively as (1) b = − det ( A , B ) Δ x The presence of det(A,B) suggests a geometrical interpretation, but I couldn't think of one. This reminds me of Cramer's Rule, but I couldn't make that connection explicit either.

Question

Suppose a line goes through the points   A  =  (      x    1    ,      y    1    ) and   B  =  (      x    2    ,      y    2    ). One can easily check (as I did today while doodling) that  b  =  −                    x        1                    y        2            −              x        2                    y        1                            x        2            −              x        1            where   b is the y-coordinate of the   y-intercept. This can be written more suggestively as                    (1)                    b        =        −                              det            (            A            ,            B            )                                Δ            x                              The presence of det(A,B) suggests a geometrical interpretation, but I couldn&#039;t think of one. This reminds me of Cramer&#039;s Rule, but I couldn&#039;t make that connection explicit either.

Josie123 · Accepted Answer

Basically you are using the fact that the area enclosed by three collinear points is zero  A  (      x    1    ,      y    1    )  ,  B  (      x    2    ,      y    x    )  ,  C  (  0  ,  b  ) are the three points, then      |                                        x            1                                                y            1                                    1                                                  x            2                                                y            2                                    1                                      0                          b                          1                      |    =  0Now expand along the bottom row to get the equation you have

Suppose a line goes through the points A = ( x 1 </msub> , y

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