Hello! I have one problem which seems not so difficult: -Find the equation of line which passes throught the point M(1,0,7), parallel of the plane 3x-y+2z-15=0 and it intersects the line \(\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}\) The equation of the line will be: \(\frac{x-1}{a_1}=\frac{y}{a_2}=\frac{z-7}{a_3}\) So we need to find \(\vec{a}(a_1,a_2,a_3)\) and we need three conditions in the system. The first condition is \(\vec{a} \circ \vec{n}=0\) or \((a_1,a_2,a_3)(3,-1,2)=0\) or \(3a_1-a_2+2a_3=0\). The second condition is the intersection of two lines, and it is: \(-17a_1+28a_2+12a_3=0\) What about the third condition?
You don't need a third condition, since the a's are defined up to a constant multiplier. To solve, simply set one of them to 1 and solve for the other 2. As you wrote the original equation, none of the a's should be 0.