Pure 9: Vectors Flashcards
Vector Equation of a Straight Line - position vector and parallel
r = a + λb
Vector Equation of a Straight Line - two points
r = c + λ(d-c)
Cartesian Equation of a Line
If a = (a1 a2 a3) and b = (b1 b2 b3), r = a + λb:
x - a1)/b1 = (y-a2)/b2 = (z-a3)/b3 (= λ
Vector Equation of a Plane
r = a + λb + μc
Cartesian Equation of a Plane
Where (a b c) is the normal:
ax + by +cz = d
(Find d by substituting a known point for x, y and z)
What is the scalar product?
a.b = a1b1 + a2b2 + a3b3 = |a||b| cosθ
When are non-zero vectors perpendicular?
When a.b = 0
When are two vectors parallel? (2)
When a.b = |a||b|
In particular a.a = |a|^2
What is the acute angle θ between two intersecting straight lines?
cosθ = | (a.b) / ( |a||b| ) |
Scalar Product Form of the Equation of a Plane (2)
r.n = a.n
Where n is the normal and a is any position vector of a point in the plane
a.n is constant
What is the acute angle θ between a line and plane?
Where r = a + λb and r.n = k:
sinθ = | (b.n) / ( |b||n| ) |
What is the acute angle θ between two planes?
Where r.n1 = k1 and r.n2 = k2:
cosθ = | (n1.n2) / ( |n1||n2| ) |
When are two lines skew? (2)
Not parallel
Do not intersect
