Vectors and planes Flashcards
(12 cards)
Scalar product of 2 vectors
a1b1 + a2b2 + a3b3
Modulus of a vector
root of a2 + b2 + c2
angle between two vectors
a.b = modamodb cos θ
so θ = cos-1 a.b/mod a x mod b
both vectors must point to the same point or both away
perpendicular vectors
dot product = 0
vectors perpendicular to 2 vectors
dot product both to 0 to make 2 simultaneous equations then sub in z = 1 as scale can be whatever
equations of 3d lines
r = a + λc where λ can be any multiple, c is a direction vector between 2 points of the line and a is any point on the line
if a is x,y,z and c is a,b,c then can be written as x-x/a = y-y/b = z-z/c
perpendicular lines
the direction vectors (lambda coefficients) will dot product = 0
equation of a plane
find a normal vector n and any point A on the plane,
then n1x + n2y + n3z = n1a + n2b + n3c (i.e dot product)
so ax + by + cz = d or move d to other side and equate to 0
finding normal vector given an equation
if ax + by + cz + d = 0, then a normal vector is a,b,c
angle between two planes
find the angle between the normal vectors of both
finding plane equation given 3 points
use 3 points to get 2 on plane vectors
then use simultaneous equations to find a normal to both
then sub into usual r.n - a.n = 0
Investigating 3 plane arrangement
turn the 3 plain equations into a 3x3 matrix
if det = 0 then there is one unique intersection
otherwise :
if infinite solutions on calc then they form a sheaf (i.e a line of solutions) or 2 planes are the same (if fully multiples including d value)
if 2 planes abc (but not d) multiples then 2 parallel and one cuts through, if all 3 multiples then all 3 parallel
if no multiples then they form a triangular prism
