2 - 4 - lines + planes in space

Question 1

vector eq of a line

Answer 1

r = a + λb
if a is the position vector and b is the direction vector

Question 2

cartesian eq of a line

Answer 2

x-a1/b1 = y-a2/b2 = z-a3/b3
if a is the position vector and b is the direction vector

Question 3

vector eq of a plane

Answer 3

r = a + λb + μc
if a is the position vector and b is the direction vector
OR
r.n = p
where n is normal vector and p = a.n

Question 4

a plane is uniquely determined by

Answer 4

• 3 points not on the same line
• a line and a point outside the line
• 2 intersecting lines

Question 5

cartesian eq of a plane

Answer 5

n1x + n2y + n3z = p
where n is normal vector and p = a.n

Question 6

line and line intersection

Answer 6

make 2 vector equations =
solve simultaneously using 2
check 3rd - if dont meet skew

Question 7

line and plane intersections

Answer 7

r = a + λb and r.n = p
dot the line eq and n and make = p
(sub in r and solve)

Question 8

line and line angle between

Answer 8

a.b/|a||b| = cosθ
a and b are the 2 direction vectors

Question 9

line and plane angle between

Answer 9

cos ⍺ = b.n / |b||n|
θ = 90 - ⍺
where b is the direction vector of the line and n is the normal vector

Question 10

plane and plane angle between

Answer 10

cos ⍺ = n1.n2/ |n1||n2|
θ = 180 - ⍺

Question 11

point and line distance between

Answer 11

D = |ax1+ by1 -c|/ √a^2 + b^2
where the point is (x1,y1)
and the line is ax + by = c

Question 12

point and plane distance between

Answer 12

D = |b.n - p|/|n|
where b is the position vector of the point and n is the normal of the plane

Question 13

skew lines distance between

Answer 13

D = |(b-a).n|/|n|
where n is normal and a,b are 2 direction vectors

Question 14

2 parallel lines distance between

Answer 14

cos θ = (b-a).d/|b-a||d|
D = |b-a|sinθ
not in formula book