6.2 - Eqx of lines in R3

Question 1

Theory: Parametrize the line ax + by = c

Answer 1

a != 0
parameter is y = t (can also be x =t)
ax = c - by
x = c/a - b/a(t)
(x, y) = (c/a, 0) + t(-b/a, 1)

Question 2

Def: What is a line

Answer 2

All variables can be expressed using 1 parameter, so dimensionally the equations is 1.

Question 3

Def: parametric eqx of line

Answer 3

x = c/a - b/a(t)
y = t

Question 4

Theory: Vector parallel to line ax + by = c if
1) Equation is given
2) A known point and equation are given.

Answer 4

Eqx of line is y = a/b(x) + c\
slope is a/b
Therefore is v = k(b,a) if both do not equal 0 or k != 0.

Use known point and line to get second random point. Find vector between them
PQ = kv
kv is the scalar multiple of line PQ as is therefore parallel.

Question 5

Def: Direction vector of line L.

Answer 5

v // L
v = (a b c)

Question 6

Def: Parametric eqx in R3.

Answer 6

x = xnot + at
y = ynot + bt
z = znot + ct

Question 7

Def: Symmetric form

Answer 7

t = t = t
x - xnot / a = y - ynot / b = z - znot / c

Question 8

Theory: What are the two things needed to find para eqx of line?

Answer 8

1. A point on the line
2. Vector parallel to the line

Question 9

Def: Distance btw line and point

Answer 9

D = norm(PQ x proj-v-PQ) = norm(PQ x v) / norm(v)
where v is a direction vector
Q is the point and P is another points on the line.