Solve Vector Intersection: A,B,C,D | Point P & Perpendicular Line

[bruceflea]

It's seems like such a basic question but I can't for the life of me remember how to do it.Given the points A(0,0,1), B(2,3,2), C(1,0,0) and D(2,2,1), verify that the line through A and B and the line through C and D intersect and find the point of intersection P. Find the equation of the line through P which is perpendicular to both these lines.I've worked out the point P to be (4,6,3) and I know that for a line to be perpendicular, the dot products of two vectors have to be 0.

I know that the equation between A and B is r = k + P(2i+3j+k) and between C and D r = i + Q(i+2j+k).

Am I right in saying that the vector of AB is 2i+3j+k and the vector of CD is i+2j+k?The answer I worked it out to be is r = 4i+6j+3k + S(-i+j-k)

 

[t!m]

Looks good to me. While the dot product part is true, an easier way to find the perpendicular vector would be the cross product, which gives a third vector perpendicular to both given vectors. CD x AB gives -i+j-k for the direction vector, as you found.

 

[vaishakh]

Okay tim provided you the important help. Anyway what Tim found was a vector perpendicular to two vectors. what you need is equation. You understand the slope of the line from cross product and write the equation carefully. For this ifthe vector is ai + bj + ck and should be passing through a spacely point x1,y1,z1, then the equation must be bc(x1 -x) = ac(y1 - y) = bc(z1 -z). verify whether this is correct.