Vector geometry
-Points, lines etc in ℝ2, ℝ3
Vectors
Vectors are quantities have a magnitude (size) and a direction . (compared to scalar with just
magnitude)
Commonly vectors are 2 or 3 dimensional.
2D: velocity over ground
3D: spatial velocity (planes, submarines), gravity
Vector represented as a tuple of numbers:
2D: as a pair of numbers ( 2 tuple) ℝ2 = {(x, y)|x, y ∈ ℝ}
⃗⃗⃗⃗⃗ = (2,1)’
P(2,1), 𝑂𝑃
3D: ℝ3 = {(x, y, z)|x, y, z ∈ ℝ}
Magnitude of a vector
Magnitude of 𝐚 = (a1 , a2 )
|𝒂| = √(a21 + a22 )
Magnitude of 𝐚 = (a1 , a2, a3 )
|𝒂| = √(a21 , +a22 + a23 )
⃗⃗⃗⃗⃗ | = |𝒒 − 𝒑|
Magnitude of |𝑃𝑄
Properties Of Vector geometry
1)vector algebra:
Equality: a=b, same direction and magititude. (a1 , a2, a3 ) = (b1 , b2, b3 )
Addition is commutative
Zero Vector: 0=(0,0,0) a+0=a
Scalar multiplication ka=(ka1 , ka2,k a3 )
Parallelism: a∥b iff a=kb
𝒂
Unit vector, vector of magnitude 1: |𝒂| is a unit vector in the direction of a
Standard Bases
i,
j,
k
Triangle inequality: |𝑎 + 𝑏| ≤ |𝑎| + |𝑏| equal iff a is parallel to b
Midpoint
let m be the midpoint of ⃗⃗⃗⃗⃗
𝐴𝐵
⃗⃗⃗⃗⃗⃗ =
𝑂𝑀
̅̅̅̅ = {ka_(lk)b|0≤k≤1} CHECK THIS
𝐴𝐵
𝒂+𝒃
2
Do 2.1 q10
Therom 2.1
If A and B are parrell (collinear) then there exists scalars l and g (not both that are zero ) st
lA+gB=0
if a and B are not parallel then l=0 and g=0
If A and B and C are coplanar then there exists scalars l and g and h (not all that are zero )
st lA+gB+hC=0
if a and B are not coplanar then l=0 and g=0 and h=0
Line
Line can be represented by any 2 points.
Vector eqation of a line
⃗⃗⃗⃗⃗ + 𝜆𝐴𝐵
⃗⃗⃗⃗⃗ }
𝐿 = {𝑃|𝑂𝐴
⃗⃗⃗⃗⃗ is the direction
𝜆𝐴𝐵
⃗⃗⃗⃗⃗
𝑂𝐴 is any point on the line.
Parametric equation of a line:
𝐿 = {(𝑥, 𝑦, 𝑧)|(𝑥, 𝑦, 𝑧) = (𝑎1 , 𝑎2 , 𝑎3 ) + 𝜆(𝑢1 , 𝑢2 , 𝑢3 )}
Cartesian equation of a line:
𝑥 = 𝑎1 + 𝜆𝑢1
𝑦 = 𝑎2 + 𝜆𝑢2
𝑧 = 𝑎3 + 𝜆𝑢3
When all parts of that are equal (ah screw it i’ll type it latter)
𝑥 − 𝑎1 𝑦 − 𝑎2 𝑧 − 𝑎3
=
=
𝑢1
𝑢2
𝑢3
Plane
A plane can specified by 3 non-collinear points
Π is a plane containing,A,B,C
To be on the plane Π you need to be a mulitiple of AB, + a multiple of AC
Vector definition of a plane.
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
Π = {𝑃|𝑂𝑃
𝑂𝐴 + 𝜆1 ⃗⃗⃗⃗⃗
𝐴𝐵 + 𝜆3 ⃗⃗⃗⃗⃗
𝐴𝐶}
Dot Products
𝑎 . 𝑏 = 𝑎1 𝑏1 + 𝑎3 𝑏3 = 𝑎𝑏 𝑇
Properties
- a.b=b.a commutative
- distributive over addition, (a+b)+c=a+(b+c)
- (λa).b=λ(a.b) Scalrmultiplcation.
-|𝑎|2 = 𝑎. 𝑎
If we have vector a and b at andle θ between them.
𝐚. 𝐛 = |𝐚||𝐛| 𝐜𝐨𝐬 𝛉
-if 2 vectors are orthogonal (at right angles) their dot product =0.
Projection
we have vector b, and vector a which is not collinear to b.
We can view vector a as being composted of some part that is parallel to b, and some part
that are orthogonal to b.
This is called the projection of a on to (or in the direction of) b.
P is the projection of a on to b. (only the part that is NOT orthogonal)
We can use this to find how close a line comes to a point.
Dot products to define a plane
We can define a plane by giving a point on it, and a normal vector to it.
take A a point on the plane. And n the normal vector to the plain. And P a point that might be
on the plain.
⃗⃗⃗⃗⃗ .n=0)
P is on the plain if (𝐴𝑃
Cross product
𝑎 = (𝑎1 , 𝑎2 , 𝑎3 )
𝑏 = (𝑏1 , 𝑏2 , 𝑏3 )
𝑎 × 𝑏 = (𝑎2 𝑏3 − 𝑎3 𝑏2 , 𝑎3 𝑏1 − 𝑎1 r d, 𝑎1 𝑏2 − 𝑎2 𝑏1 )
𝑎 × 𝑏 is normal to both a and b.
We determine the direction (up or down as relivitve determined by the plane defined by a, b (eg
OA,OB plane).
How to
𝑎 = (𝑎1 , 𝑎2 , 𝑎3 )
𝑏 = (𝑏1 , 𝑏2 , 𝑏3 )
i
j
k
i
j
a1 a2 a3 a1 a2
b1 b2 b3 b1 b2
Top-right to bottom left diagonals contribute negative terms
top left to bottom right contribute positive terms.
So we have:
+i.a1.b2
+j.a3b1
+k.a1.b2
-j.a1.b3 ↷
-i.a3.b2 ↝
-k.a2.b1
Plane define by cross product.
if we have a plane with points on it A,B and C; AB=a, AC=b.
And a point that can be on the plane P, position AP
If it is on the plane then: ⃗⃗⃗⃗⃗
𝐴𝑃 ∙ (𝑎̃ × 𝑏̃)=0
Properties of the cross productat
Looing at vactors a, b . c
̃ × 𝑏̃ = −(𝑏̃ × 𝑎)
̃
Not communtiive ′𝑎
̃ × 𝑏̃)×c≠𝑐 × (𝑎̃ × 𝑏̃)
Not accosicative (𝑎
̃
Is distributive (𝑎 + 𝑏̃)×c=𝑎̃ × 𝑐 + 𝑏̃×c
𝑎̃ × 𝑎̃ = 0
Distence of a point from a plane
⃗⃗⃗⃗⃗
𝐴𝑃 ∙𝒏
MP=𝑃𝑟𝑜𝑗𝒏 ⃗⃗⃗⃗⃗
𝐴𝑃 = ( 𝒏∙𝒏 ) 𝒏
this projection on to the normal (n), of Point P (looking from any point on the plane, A),
Shows the dististence, when taken the maginitude.
M is the closest point on the plane to the point P.
Equation of a plane:
⃗⃗⃗⃗⃗ ∙ 𝑛̃ = 0
equation of a plane is 𝐴𝑃
let 𝑛̃ = (𝑛1 , 𝑛2 , 𝑛3 )
⃗⃗⃗⃗⃗
𝐴𝑃 = (𝑥 − 𝑎, 𝑦 − 𝑏, 𝑧 − 𝑐)
⃗⃗⃗⃗⃗
𝐴𝑃 ∙ 𝑛̃ = 0
(𝑥 − 𝑎, 𝑦 − 𝑏, 𝑧 − 𝑐) ∙ (𝑛1 , 𝑛2 , 𝑛3 ) = 0
𝑛1 𝑥 + 𝑛2 𝑦 + 𝑛3 𝑧 = 𝑛1 𝑎 + 𝑛2 𝑏 + 𝑛3 𝑐
Which is the equation of the plane.