# Need help visualising line orthogonality to a plane. Why a line cannot be orthogonal to 2 parallel planes but a 3rd plane can.

I am just having a difficult time processing this. Suppose I have a line which is orthogonal to a plane meaning that it is parallel to the normal vector of the plane. I just don't understand why a line cannot be orthogonal to two non-parallel planes while a 3rd plane can be orthogonal. Can the line not simply point the same direction as to the normal vector of the 3rd plane which is orthogonal to the 2 non-parallel planes. Like for example, one of the questions I got was: Find $$P_3$$ if it is possible between 2 planes with equations:

$$P_1 : −2x + y − 4z = 2$$,

$$P_2 : x + 2y = 7$$.

I simply found the normal vectors of $$(-2,1,-4)$$ and $$(1,2,0)$$ and made normal vector of $$P_3$$ be $$(x,y,z)$$ with the dot product $$n_3 . n_1 = 0$$ and $$n_2 . n_1 = 0$$ Used matrices, got 1 free variable and got $$x = -2t$$, $$y = t$$ , and $$z = 5t/4$$ where t can be any real number and found a possible general equation equating $$t = 4$$ to get $$P_3 = -8x + 4y + 5z$$ from normal vector of $$P_3 (-8, 4, 5)$$

Find a line which is orthogonal to both $$P_1$$ and $$P_2$$. Give an equation if it is possible and explain why it is or is not possible: I simply don't know how to explain why a line is or is not possible though or visualize this. So I need help understanding how this works.

Any help is appreciated. Thanks.

• What definitions are you using? A line is orthogonal to a plane if it is parallel to the normal vector, i.e., if it is orthogonal to any vector in the plane. – Elliot G Sep 21 at 4:50
• sorry thats what I meant, I wrote it wrong. – Ese material Mexico taco Sep 21 at 5:40

A line, if orthogonal to two planes, must be parallel to both normal vectors of the two planes. When you set $$n_1 \cdot n_3 = 0, n_2 \cdot n_3 = 0$$ this means that the line spanned by $$n_3$$ is perpendicular to both $$n_1$$ and $$n_2.$$ But you want parallel. If you think about it, a line cannot be parallel to simultaneously to two intersecting vectors. That is why you need a plane - specifically a plane spanned by the normals to the two intersecting planes.