I have two lines defined by a starting point ( the line passes this point...) and a direction vector. $$p_1=(10,0,0)$$$$v_1=2i+4j+3k$$$$p_2=(310,480,400)$$$$v_1=-4i-6j-5k$$
I need to ( 1 ) show whether or not these lines intersect, and if they do ( 2 ) what is the point of intersection.
Solution : Since these lines are not parallel on the basis of the direction vectors, we can say that they do indeed intersect. In order to find the point of intersection, we must first write these lines in vector parametric form - which gives us a system of equations
$$10+2t=310-4s $$$$4t=480-6s $$$$3t=400-5s $$
From here, we just solve the parameters $s$ and $t$ and use them to find the co-ordinates of our intersection point. Solving for s and t, however, I get $s = 60$ and $t = 30$ which satisfies the first two equations, but no the third one. Is there an error in calculation or does this say something about our problem?