Hello Schdero Originally Posted by

**Schdero** This seems to be a logical solution, although a little bit above my level of mathematics, as I do not see how to solve one equation with 4 unknowns...

It's very straightforward. If you've reached vector products, you must have seen something like this before.

The expressions on either side of the $\displaystyle \equiv$ sign are identical, so we can compare coefficients of $\displaystyle x, y, z$ and constant, to get: $\displaystyle 4\lambda+\mu=1$ ...(1)

$\displaystyle 3\lambda + 2\mu = b$ ...(2)

$\displaystyle 2\lambda+3\mu = c$ ...(3)

$\displaystyle \lambda + 4\mu = 7$ ...(4)

Multiply (1) by 4, and subtract (4):$\displaystyle 15\lambda = -3$

$\displaystyle \Rightarrow \lambda = -\tfrac15$

... etc.

But what ails me more than the actual way of solving this problem, is the following:

Be the normal vector of P1-> (nP1),

the n. vector of P2 ->(nP2)

the n. vector of p3 -> (nP3)

and Direction vector of the line of intersection ->(d)

(nP1)x(nP2) = (d)

Thus: (nP1)/(nP2)x(nP3) = t(d)

Even though the exact position of the line is not specified, there is only ONE direction vector of the line. So why does (nP2)x(nP3)=t(d) not lead to the correct solution but only to a correlation between b and c?

As *running-gag* explained, you have found the direction vector of the line of intersection to be$\displaystyle \begin{pmatrix}5\\-10\\5\end{pmatrix}$

and the relation between $\displaystyle b$ and $\displaystyle c$ to be$\displaystyle c = 2b-1$

What you have found so far, then, is a family of planes of the form$\displaystyle x + by + (2b-1)z + 7 = 0$

whose property is that their normal is perpendicular to the vector$\displaystyle \begin{pmatrix}5\\-10\\5\end{pmatrix}$

which can be demonstrated by forming the scalar (dot) product of the normal with this vector:$\displaystyle \begin{pmatrix}1\\b\\2b-1\end{pmatrix}\cdot\begin{pmatrix}5\\-10\\5\end{pmatrix}=(1)(5)+b(-10)+(2b-1)(5)$ For any value of $\displaystyle b$, then, the plane satisfies all the conditions you have written down. You need to find an additional condition that specifies which particular plane contains the actual line of intersection of P1 and P2.

I hope that may make it clearer.

Grandad