diff --git a/docs/mcv4u7.md b/docs/mcv4u7.md index e069b16..f42a501 100644 --- a/docs/mcv4u7.md +++ b/docs/mcv4u7.md @@ -615,6 +615,26 @@ An initial point vector can be solved by setting any of the variables ($x,y,z$) The **angle between two planes** is equal to the angle between their normal direction vectors, which can be determined using the dot product formula. +When looking at three planes: + +If all three normals are scalar multiples: + + - If all three $D$-values are scalar multiples, the planes are parallel and coincident and they have infinite points of intersection along the plane equation. + - Otherwise, there are no solutions and the planes are parallel and distinct and/or parallel and coincident for two. + +If two normals are scalar multiples: + + - If the two parallel planes are coincident with the same $D$-values, there will be a line of intersection much like solving for intersection between two planes. + - Otherwise, the two parallel planes are distinct, forming a Z-pattern with the third plane and so there is no solution. + +If no normals are scalar multiples: + + - If the triple scalar product of the three planes is equal to zero, the normal vectors are not coplanar and so there will be a point of intersection. + - Alternatively, by solving the scalar equations for the planes, if: + - the result is a contradiction (e.g., $0 = 3$), there is no solution + - the result is true with no variable (e.g., $0 = 0$), there are is an infinite number of solutions along a line + - the result contains a variable (e.g., $t = 4$), there is a single point of intersection at the parameter $t$. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf) diff --git a/mkdocs.yml b/mkdocs.yml index 8cc235d..a42e1a5 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -1,4 +1,5 @@ site_name: Eifueo +site_url: "" nav: - Home: index.md - SL Physics 1: sph3u7.md