6.3 - Eqx of Planes in R3

Question 1

Def: Scalar eqx of plane, value of d, and alternative form.

Answer 1

ax + by + cz = d
d = axnot + bynot + cznot = normal vector*xnot
a(x - xnot) + b(y - ynot) + c(z - znot) = 0

Question 2

Def: normal vector

Answer 2

n = (a b c)
perpinduclar to the plane

Question 3

Def: Direction Vector

Answer 3

v is a dir vector

Question 4

Theory: If a dir vector v is intersection of two planes, v = n1 x n2 of both planes.

Question 5

Theory: What 2 things are needed to determine eqx of a plane?

Answer 5

1. Point in plane
2. Normal vector to plane
   - Can be found using cross product of 2 vectors in plane

Question 6

Theory: What are the steps to find the equation of a plane containing a point and a line of intersection btw two planes.

Answer 6

1) Find dir vector which is the first vector on plane
- Using normal vector from planes
2) Find point on plane and w, another vector on plane
- Point on plane can be found by solving system of two planes
- w is the vector between known point and calculated point
3) Find normal and solve
- Normal from two vectors
- Use point and normal to make eqx

Question 7

Theory: What is distance between points and something?

Answer 7

min|PQ|, where Q is the point and P is the plane/line. This is the minimum distance.

Question 8

Def: Distance formula btw plane and point.

Answer 8

dist = | ax1 + by1 + cz1 - d| / sqrt(a^2 + b^2 + c^2)
where (x1, y1, z1) being the point, and abc being normal vector of plane.