Interphase Calc - Week 1

R²

All pairs of coordinates in 2D euclidian space

R³

set of all triple coordinates in 3D euclidian space

In nth Dimension

v = (v₁,… vₙ) = (vi); i∈[n] (for i being in the set of n)

vectors

not bound to origin

vector between 2 points

AB - (b₁ - a₁, b₂ - a₂)… end point minus start point

Addition and subtraction

Tip to tail, add each component

Canonical Basis of R^n

ei = unit displacement along ith coordinate

ei = (0,0,…,0,1,…,0)

v = v₁e₁+…+vnen = ∑ i=1 → n eiei

(for ex: e1 in 3D is (1,0,0) or ihat)

Norm of vector

magnitude, √∑(xi)²

Scalar of magnitude

||λv|| = λ||v||

Unit Vector

vector with magnitude 1… vhat = v/||v||

Inner Product

\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_{i} v_{i} , gives scalar output

inner product cosine

u ⋅ v = ||u|| ||v|| cos(θ)

Relation fo dot product to theta

u ⋅ v > 0 when theta is acute, 0 when theta is obtuse, =0 when they form a right angle

properties of inner product

-distributes over addition

-can extract scalars

-has symmetry u ⋅ v = v ⋅ u

u ⋅ u

||u||²

||u+v||²

||u||²+||v||²+2u ⋅ v

Cross product

outputs the vector that is perpendicular to both u and v

||u x v||

||u|| ||v|| sinθ

Properties of cross product

Antisymmetric u x v = -v x u

Distributes over addition (a+b) x v = a x v + b x v

pull out scalars

Equation of plane

AP ⋅ n (normal vector to plane) = 0

ax+by+cz = d

Scalar component of v in the direction of u

(u ⋅ v) / ||u||