## The Point of Intersection of Two Lines in Three Dimensions

If two lines intersect, theyare both in the same place at the same time, so to speak. We don'tknow what the point is, but because they both meet at the same point,we can put the equations of the lines equal to each other. This willresult in simultaneous equations for the parameters. We solve them,then substitute back in to the lines to find the point.

Example: and Findif the lines intersect and if they do intersect, find the coordinatesof the point of intersection.

Put Put each component of equalto the corresponding component of Weobtain (1) (2) (3)

(2)-(3) gives Substitute into(1), (2) or (3) to give t=1. and Hence both equations meet atthe same point when If the lines do notintersect at a point then the two points will not be the same orequivalently there will not be values of s ant t that satisfy allthree equations simultaneously.

Example: and We form the same equationsas before and perform (2)-(3) to get butequation (1) is now and donot satisfy this equation so these lines do not intersect. 