1. Parallel and distinct
2. Parallel and coincident
3. Intersecting (not parallel)
4. Skew (not parallel, not intersecting)
How many possible cases are there for relative positions of two lines? And what are they?
100
line lies on plane therefore there are infinite solutions
Determine if the following intersects:
π2: x + 3y -4z = 10
L2: x = 4+6t
y = -7+2t
z = 1+3t
100
1. intersecting (into a line)
2. Coincident
Draw out the two possible relative planes
100
You use the triple scalar product n1xn2xn3
In other words,
If n1xn2xn3=0, then the three planes are coplanar
If n1xn2xn3 does not equal zero, then the three planes are not coplanar
How do you check if normals are Coplanar?
100
d = |Ax1 + By1 + C| / √(A^2 + B^2)
What is the equation for 'Distance from a point to a line in R^2'
200
Case 2, infinite many solutions
Classify if the following has zero, one, or infinite many solutions
L1: 4x-6y=-10
L2: 6x-9y=-15
200
Yes at (3,0,-1)
Determine if the Following intersect:
π1: 9x + 13y - 2z =29
L2: x = 5+2t
y = -5-5t
z = 2+3t
200
proportional
If the planes are parallel, the the coefficients A,B,C are ________
200
Answers will be provided in package
Sketch the 4 possible Consistent/Compatible system of equations cases
200
a) Find a specific point on one of these parallel lines.
b) Find the distance from that specific point to the other line using one of the relations above
How do you find the distance between two parallel lines?
300
zero solutions, case 3
Classify the following as having zero, one, or infinite many solutions
L1: [x,y] = [1,5] + s[-6,8]
L2: [x,y] = [2,1] + t[9,-12]
300
since every t satisfies the eon the line lies on the plane and there are infinite many solutions
Determine if the following intersects:
π3:4x-y+1z=-1
L3: [x,y,z] = [-2,4,1]+ t[3,1,-1]
300
proportional
Since we know that the planes are coincident, the coefficients of A,B,C,D must be ________
300
Answers will be provided in package
Sketch the 4 possible Inconsistent/Incompatible system of equations cases
300
a) Find a specific point into one of these planes
b) Find the distance between that specific point and the other plane using one of the formulas above.
How do you find the distance from a point to a plane
400
lines parallel and distinct, no solutions, case 3
Determine if these lines intersect:
L1: [x,y,z] = [5,11,2] + s[1,5,-2]
L2: [x,y,z] = [1,-9,9] + t[1,5,-2]
400
single point, infinite points, no points
Draw out the three possible cases of relative position of a line and a plane
400
A,B,C are proportional, therefore planes are coincident.
Classify the pair of planes as distinct, coincident, or intersecting. Do so without solving.
π1: 2x-3y+z-1=0
π2: 4x-6y+2z-2=0
400
l: (1,-3,0) + T(-4+2)
Find the intersection of the planes:
x+y+2z=-2
3x-y+14z=6
x+2y=-5
400
d = |Ax1 + BY1 + Cz1 +D| / √(A^2 + B^2 + C^2)
What is the equation for 'Distance from a point to a plane'?
500
one unique solution (case 1) POI at (1,-1,3)
Determine how the lines intersect. If there is a solution, find the point of intersection.
L1: [x,y,z] = [7,2,-6] + s[2,1,-3]
L2: [x,y,z] = [3,9,13] + t[1,5,5]
500
1. substitute the parametric eons of a line, into the cartesian equation of the plane
2. solve (if possible) the equation for the parameter t.
3. Substitute the value of the parameter t into the parametric eqn of the line too get the point of intersection
Write out the steps needed to get the intersection between a line L and a plane π
500
compatible means there is at least one solutions, incompatible means there are no solutions.
Describe the difference between between a system of equations that are compatible vs. a system of equations that are incompatible
500
n1=(2,-1,3)
n2=(4,-2,6)
n3=(6,-3,9)
n2=2n1
n3=3n1
State the normal vectors for each plane
π1:2x-y+3z-2=0
π2:4x-2y+6z-3=0
π3:6x-3y+9z-4=0
500
Distance = 6.8
Find the distance from the point P=(4,−4,3) to the plane
2x−2y+5z+8=0