calc ch 8

Question 1

what are the 4 possible intersections with lines

Answer 1

1. distinct, intersect - 1 solution
2. skew - 0 solutions
3. coincident - ∞ solutions
4. parallel, distinct - 0 solutions

skew: disctint & dont intersect
coincident: parallel & intersect

Question 2

how do you check if lines are parallel

Answer 2

m₁=km₂

is parallel

Question 3

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]

Answer 3

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

Question 4

how to find if distinct lines are skew or 1 POI

Answer 4

turn everything into parametric equations, make both sides equal eachother, solve for t and s.

if LS = RS -> POI
if LS ≠ RS -> skew

Question 5

how to find POI if the lines are distinct

Answer 5

take the values of s & t and sub into any origional parametric equation, solving for x,y,z.
POI = (x,y,z)

Question 6

what is the shortest distance formula

Answer 6

|projP1P2n| = |P1P2*n/|n||

Question 7

what are the 3 possibilities with intersections lines and planes

Answer 7

• plane and line intersect - 1 intersection
• line lies on the plane - ∞ solutions
• line is parallel with the plane - 0 solutions

Question 8

what is the formula to check if a line is parallel with a plane

Answer 8

m * n = 0
parallel

Question 9

how to check if coincide when plane and line are parallel

Answer 9

sub Po from the line into plane equation

Question 10

how to find the POI of line and plane if distinct

Answer 10

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

Question 11

how to find the shortest distance between a line and a plane

Answer 11

you need 2 points, create a point that lays on the plane then find P1P2 etc.

Question 12

shortest distance between a point and a plane

Answer 12

• check if the point is on the plane (should equal 0)
• create a second point and do the same

Question 13

what are the 3 possible intersections of two planes

Answer 13

1. planes are distinct - 1line of intersection} n ≠ kn
2. planes are parallel, intersect - ∞ solutions} n = kn, π = kπ
3. planes are parallel, distinct - 0 solutons} n = kn, π ≠ kπ

Question 14

how to find the solution to two planes

n * m = 0

Answer 14

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

Question 15

ex to try:

π1 = 3x - 2y + z = 4
π2 = 6x - 4y +3z = 7

how to find the solution to two planes

n * m ≠ 0

Answer 15

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]

Question 16

what are the 3 possibilities of intersection with three planes

Answer 16

• 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

Question 17

what is the ‘line or POI’ formula

Answer 17

n₁(n₂ x n₃)≠0 -> POI
n₁(n₂ x n₃)=0 -> line

Question 18

• how to tell the 3 planes are parallel
• what it means
• what to do

Answer 18

• 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

Question 19

• how to tell if the 3 planes are distinct
• what it means
• what to do

Answer 19

• 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

Question 20

what happens if the 3 planes elimination break?

Answer 20

there are no solutions. STOP doing the question.

Question 21

there is a LOI. what happens if the equations do not equal eachother?

Answer 21

there is an inconsistent systen. STOP diong the question.