9.3 Intersection of Two Planes Flashcards
how do you tell when a system is impossible to solve?
the the coefficients of x,y and z in the cartesian equation are zero
0x+0y+0z=3
given an system of two equations where one has all coefficients of x,y and z equal to 0, how do find an equation that might lead to this solution?
find an equation and use the appropriate operation to find an answer equivalent to the impossible equation
how to determine the value of three variable in a system of two equation such that their planes are coincident?
- use one of the coefficients given and figure out what the equations are multiples of
- use the multiple and the given numbers to find out what the variables are
how to determine the value of three variables in a system of two equations such that the planes are parallel and not coincident?
- figure out what the equations are multiples of
- find values of the direction vector that is are multiples of each other
- find values of the constant that are not parallel
how to find the value of a variable beside x,y or z such that two planes intersect at right angles?
- the direction vectors of both planes must have a product of 0, so multiply the normals of each plane together
- isolate for the variable
solve the system of equations meaning?
find values of x,y and z that make both equations true at the same time
how to figure out if two planes intersect?
- state the normal of both planes
- state the D values of both planes
- see if the normals are scalar multiples of one another, if they are, then the planes are parallel, if they are not, then the lines intersect`
- see if the D values are scalar multiples of one another, if they are, then the lines are coincident and they have infinite amount of solutions
what does it mean if two planes have normal vectors that are multiples of one another but D values that are not
this means that planes are parallel but do not intersect
how to find the value of a variable in a system of two equation to have an infinite number of solutions?
- find what the two equations are multiples of
- use the coefficient given and the multiple to find the values of the variable
how to find the value of a variable beside x,y or z and is a constant in a system in one of the systems of two equations such that the equations have no solutions?
there would be no value to the variable. The k in the direction vector must be a multiple to the other plane but the constant must be a different number in order for the planes to intersect
how to determine the vector equation of a line that passes through a point and is parallel to the line of intersection of two planes?
- isolate for one of the variables
- substitute the equation for the variable into the other equation to find the equation of the other variable
- set the variable left equal to t
- rewrite the parametric equation in terms of t instead of x, y or z
- write a vector equation in terms of the direction vector from the parametric equations
- write the final vector equation with the point given and the direction vector t
how to show that the line of intersection between two planes lies on a plane? (three planes in total)
find the line of intersection between the two planes using substitution
- Add the two equations to create another equation.
- Isolate for variable in third equation
- substitute third equation into first equation
- Remaining variable equal t
- Rewrite parametric equations
- substitute the parametric equations into the plane you want to prove the line lies on
- if both sides of the equation equals zero, then the line of intersection lies on the plane
how to find the parametric equations of two planes?
- isolate for one variable
- substitute the equation of the variable into the other equation to find the equation to the other variable
- set the variable left equal to t
- rewrite the parametric equations in terms of t
how to find the cartesian equation of a plane parallel to the line with equation x=-2y=3z and contains the line of intersection between two planes?
- set all the variables equal to s and isolate for the variable
- write the vector equation for the s direction vector
find the point of intersection between the two planes
- isolate for a variable
- substitute the equation of the variable into the other equation to find the equation of the other variable
- set the variable left equal to t
- rewrite the vector equation in terms of t
- find the cross product of direction vectors s and t
- find the cartesian equation using the cross product and the point given
NOTE: make sure there are not fraction in the cartesian equation
make sure the A value is positive by multiplying everything by -1
