Advanced Math/Linear algebra 1
I am having difficulty with this question, i have been trying for a couple of days with no good results. Any help would be appreciated.
What is the relationship of the given plane 2x-10y-2z+2=0 to each of the planes below? Are they coincident, parallel or intersecting?
1. 7x-35y-7z = 112
2. -3x-4y+4z = -33
3. -11x+55y+11z = 11
4. 5x+5y+1z = -5
Billy, Timmy, Sam,
Wish you had more specific questions or could at least list some of the definitions and results presented to you in your course work. The subject of lines and planes in 3-D is covered pretty succinctly in various references; one example that you should look at is
These eqns for the planes are called scalar equations. Since this eqn is derived by the dot product of a vector in the plane with the vector normal to the plane, the normal vector can be seen plain as day in the equation; it is comprised of the coefficients of x,y,z the scalar eqn. For the given eqn it is
n = (2,-10,-2).
A plane is parallel to another plane if their normal vectors are parallel. As in previous answers, we can show that 2 vectors are parallel if they (or more precisely, their components) are proportional to each other. For plane #3 compared with the given plane, we have for the normals
(2,-10,-2) = (-2/11)(-11,55,11)
which means that the normal vectors are parallel and so the planes are parallel. Note also that the constant on the right side of #3, when multiplied by -2/11, is equal to -2, which is equal to the constant in the given eqn. Thus, the planes are actually coincident (they are parallel and contain a common point).
For plane #1, we have
(2,-10,-2) = (2/7)(7,-35,-7), i.e, they are proportional (proportionality constant = 2/7)
so they are also parallel. However, the constants are not equal so the planes are not coincident.
Planes #2 and #4 are not parallel and so intersect.
Let me know how your analysis goes.