How can I convert the equation system form of the line to its corresponding vector form?

I currently have:

$x + y - z = 8$

$2x + 2y + z = 15$

I need the line description in a form of $(a_1, b_1, c_1) + \lambda (a_2, b_2, c_2)$, so that the following is true:

$(x, y, z) = (a_1 + \lambda*a_2, b_1 + \lambda*b_2, c_1 + \lambda*c_3)$

How do I do that?

  • $\begingroup$ Special Solution + $\lambda \times$ Null Space. $\endgroup$ – Inquest Oct 18 '12 at 20:51
  • $\begingroup$ Forgive me my ignorance, but could you elaborate, please? $\endgroup$ – arik-so Oct 18 '12 at 20:54

The two given equations represent planes, and the required line is their intersection. They can be written in vector form as $$(x,y,z)\cdot U = 8$$ $$(x,y,z)\cdot V = 15$$ where $U = (1,1,-1)$ and $V = (2,2,1)$ are vectors that are normal to the two planes.

It is obvious (I think) that the line is parallel to the cross product vector $U \times V$. So, we can use $(a_2, b_2, c_2) = U \times V$.

Next, we need some point $(a_1,b_1,c_1)$ on the line. There are many possible choices, of course. In effect, we just need to choose some third plane and intersect our line with this. In other words, we add a third linear equation to the two we already have, and solve. One easy choice is the plane $x=0$. Using this equation together with the original two gives the solution $(a_1,b_1,c_1)=(0,23/3,-1/3)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.