 There are several ways we can find the distance between a line and a point. One method uses the dot product trigonometry, another method uses calculus optimization approaches, and a third approach uses systems of linear equations. So again, let's consider our problem of finding the distance between a point and a line which happens to be given in vector form. So suppose we have some vector that corresponds to the distance between the point and the line. Then we can use that vector to find a point on the line, but the vector equations tell us that all points on the line satisfy for some t, and compare the coordinates give us. Now since the distance vector should be orthogonal to the direction vector of the line, then the dot product should be zero, and this gives us a system of four equations in four unknowns. So rewriting this in standard form, writing as an augmented coefficient matrix and reducing, and we have a solution. Now remember x1, x2, x3 were the actual components of our distance vector, and so our distance will be, and so we've looked at three different ways to find the distance between a point and a line in space. We can use the dot product in trigonometry, we can use calculus optimization techniques, or we can use linear algebra. Each has its advantages and disadvantages. The dot product in trigonometry only require knowing trigonometry. Calculus optimization can be applied to more general curves, and the linear algebra method can be implemented using integer arithmetic. And it should be clear that the best method is whichever method solves a problem most easily.