Find the vector equation of a plane which goes through the origin and which contains the line r=2i+λ(j+k)?

Question

I've tried to do it by using the points 0,0,0 and two other points contained in the line and creating a matrix. However, whenever i try to calculate it, I end up recieving a singular matrix error in my calculator. The points arnt collinear, why is this?

Cartesian equation is fine, it's easy to convert.

Answer 1

Try this. Get two points on the line. If lambda = 0 and 1, you have A = 2i and B = 2i + j + k.

Now the cross product vector C = A x B is perpendicular to the plane. Here, C = -2j + 2k.

Now let D = xi + yj + zk be a point in the plane you want. Then C dot D must equal zero. That is, -2y + 2z = 0, or y=z (for all x) is the equation of the plane you want.

You have to turn that into a vector equation.