1. EQUATIONS OF LINES
1.1. Equations in R2
1.1.1. Two Distinct Points OR One Point & Direction Vector
1.1.1.1. Tail at Origin and Tail at (a,b)
1.1.1.1.1. Direction Vector: m=(a,b)
1.1.2. Vector Equation
1.1.2.1. r=r0 + tm
1.1.2.2. r= (x0, y0) + t(a,b)
1.1.3. Parametric Equation
1.1.3.1. x= x0 + ta
1.1.3.2. y= y0 + tb
1.1.4. Cartesian Equation
1.1.4.1. y= mx +b
1.1.4.2. Ax + By + C = 0
1.2. Equations in R3
1.2.1. Two Distinct Points OR One Point & Direction Vector
1.2.1.1. Tail at Origin and Tail at (a,b)
1.2.1.2. Direction Vector: m=(a,b)
1.2.2. Vector Equation
1.2.2.1. r=r0 + tm
1.2.2.2. r= (x0, y0, z0) + t(a,b,c)
1.2.3. Parametric Equation
1.2.3.1. x= x0 + ta
1.2.3.2. y= y0 + tb
1.2.3.3. z= z0 + tc
1.2.4. Symmetric Equation
1.2.4.1. 𝑥−𝑥0/a = 𝑦−𝑦0/b = 𝑧−𝑧0/c
2. EQUATIIONS OF PLANES
2.1. Equations in R3
2.1.1. Four Ways
2.1.1.1. A Line and a Point Not on the Line
2.1.1.2. Two Intersecting Lines
2.1.1.3. Three Noncollinear Points
2.1.1.4. Two Parallel and Non-Coincident Lines
2.1.2. Vector Equation
2.1.2.1. r= r0 + sa + tb
2.1.2.2. r= (x0, y0, z0) + s(a1, a2, a3) + t (b1, b2, b3)
2.1.3. Parametric Equation
2.1.3.1. x= x0 + sa1 + tb1
2.1.3.2. y= y0 + sa2 + tb2
2.1.3.3. z= z0 + sa3 + tb3
2.1.4. Cartesian Equation
2.1.4.1. normal: n= (a,b,c)
2.1.4.2. Ax + By + Cz + D = 0
2.1.5. Angle Between Intersecting Planes
2.1.5.1. cosθ = n1 * n2/ |n1| |n2|
3. INTERSECTION OF LINES & PLANES
3.1. Intersection of a Line and a Plane
3.1.1. Three Ways
3.1.1.1. Case 1: One solution (intersects at one point)
3.1.1.2. Case 2: No solution (line is parallel to plane)
3.1.1.3. Case 3: Infinite solutions (line is on plane)
3.1.2. Strategies to Solve
3.1.2.1. 1. write equation of plane in cartesian form
3.1.2.2. 2. write equation of line in parametric form
3.1.2.3. 3. substitute the parametric equation into x,y,z values
3.1.2.3.1. a. t= # (one solution)
3.1.2.3.2. b. 0t= # (no solution)
3.1.2.3.3. c. 0t=0 (infinite solutions)
3.2. Intersection of Two Lines
3.2.1. Four Ways
3.2.1.1. Case 1: one solution (not parallel & intersect at one point)
3.2.1.2. Case 2: infinite solutions (lines are coincident)
3.2.1.3. Case 3: no solution (lines are parallel)
3.2.1.4. Case 4: no solution (not parallel and no skew lines)
3.2.2. Strategies to Solve
3.2.2.1. 1. Parallel of Coincident Lines
3.2.2.1.1. a. direction vectors are scalar= parallel or coincident
3.2.2.1.2. b. substitute the point on line one into second equation
3.2.2.1.3. c point on second line = coincident & point no on second like = parallel
3.2.2.2. 2. Intersect at a Point or Skew
3.2.2.2.1. a. direction vectors different= intersect at point or skew
3.2.2.2.2. b. change both lines into parametric form & set equal
3.2.2.2.3. c. use two equations to solve and plug into third
3.3. Intersection of Two Planes
3.3.1. Three Ways
3.3.1.1. Case 1: Infinite solution along line
3.3.1.2. Case 2: No solution (planes are parallel)
3.3.1.3. Case 3: Infinite solutions (planes are coincident)
3.3.2. Strategies to Solve
3.3.2.1. 1. Parallel or Coincident Planes
3.3.2.1.1. a. normal vectors parallel = planes parallel or coincident
3.3.2.1.2. b. Coincident= one equation in scalar multiple
3.3.2.2. 2. Infinite Solutions
3.3.2.2.1. a. elimination
3.3.2.2.2. b. choose 1 variable to = t
3.3.2.2.3. c. write in terms of t to get parametric equation of line