calc ch 8 Flashcards
what are the 4 possible intersections with lines
- distinct, intersect - 1 solution
- skew - 0 solutions
- coincident - ∞ solutions
- parallel, distinct - 0 solutions
skew: disctint & dont intersect
coincident: parallel & intersect
how do you check if lines are parallel
m₁=km₂
is parallel
how to find if parallel lines are distinct or coincide
Ex:
[x,y,z] = [5,-2.-8] + s[-3,2,5]
[x,y,z] = [-4,0,2] + t[-3,2,5]
substitute Po into x,y,z in parametric equaitons and solve for t
if t = all the same # -> intersect
if t ≠ all the same # -> distinct
ans:
parallel & distinct
how to find if distinct lines are skew or 1 POI
turn everything into parametric equations, make both sides equal eachother, solve for t and s.
if LS = RS -> POI
if LS ≠ RS -> skew
how to find POI if the lines are distinct
take the values of s & t and sub into any origional parametric equation, solving for x,y,z.
POI = (x,y,z)
what is the shortest distance formula
|projP1P2n| = |P1P2*n/|n||
what are the 3 possibilities with intersections lines and planes
- plane and line intersect - 1 intersection
- line lies on the plane - ∞ solutions
- line is parallel with the plane - 0 solutions
what is the formula to check if a line is parallel with a plane
m * n = 0
parallel
how to check if coincide when plane and line are parallel
sub Po from the line into plane equation
how to find the POI of line and plane if distinct
sub the x=[], y=[], z[] into equation for the plane to solve for t
then sub t into the parametric equations
the blank is what you substitiue
how to find the shortest distance between a line and a plane
you need 2 points, create a point that lays on the plane then find P1P2 etc.
shortest distance between a point and a plane
- check if the point is on the plane (should equal 0)
- create a second point and do the same
what are the 3 possible intersections of two planes
- planes are distinct - 1line of intersection} n ≠ kn
- planes are parallel, intersect - ∞ solutions} n = kn, π = kπ
- planes are parallel, distinct - 0 solutons} n = kn, π ≠ kπ
how to find the solution to two planes
n * m = 0
check to see if the planes share a point
1. create a point in the equation
2. put it into the other, if it = 0, they coincide
ex to try:
π1 = 3x - 2y + z = 4
π2 = 6x - 4y +3z = 7
how to find the solution to two planes
n * m ≠ 0
this is the little super long thing…
1. eliminate one of the variables using elimination
2. let one variable = t
3. sub t into equation and solve for the other variable
4. etc….. you know it
check to see if its right by substituting x,y,z values into an origional equation LS [will] =RS
answer: [x,y,z] = [0,-5/2,-1] + t[1,3/2,0]
what are the 3 possibilities of intersection with three planes
- planes distinct, intersect - 1 solution} n ≠ kn ≠ ln, n₁*(n₂ x n₃)≠0
- planes distinct, intersect line - ∞ solutions} n ≠ kn ≠ ln, n₁*(n₂ x n₃)=0
- planes are coincident - ∞ solutions} n = kn = ln, P1 is on π₁, π₂, and π₃
what is the ‘line or POI’ formula
n₁(n₂ x n₃)≠0 -> POI
n₁(n₂ x n₃)=0 -> line
- how to tell the 3 planes are parallel
- what it means
- what to do
- checking the normal
- there is one one possibility that the planes are parallel -> the planes are coincident.
- check if they all share the same point
- how to tell if the 3 planes are distinct
- what it means
- what to do
- 2 possibilities. if the n₁(n₂ x n₃)≠0 it means POI |&| n₁(n₂ x n₃)=0 means line
- do the elimination thing and if its a line make sure that the new equations are the same
what happens if the 3 planes elimination break?
there are no solutions. STOP doing the question.
there is a LOI. what happens if the equations do not equal eachother?
there is an inconsistent systen. STOP diong the question.
