6.1 - Inner Product, Length, and Orthogonality

For vectors:

u=((u_1),(u_2),…,(u_n)), v=((v_1),(v_2),…,(v_n))

The inner product is:

u⋅v=(u_1)*(v_1)+(u_2)*(v_2)+…+(u_n)*(v_n)

Properties

1. u⋅v=v⋅u

2. u⋅(v+w)=u⋅v+u⋅w

3. (c*u)⋅v=c*(u⋅v)

4. u⋅u=0↔u=0
   

Length (Norm) of a Vector

abs(v)=√(,v⋅v)

OR:

abs(v)=√(,(v2/1+v2/2+…+v2/n))

Unit Vector (Normalization)

To make a unit vector in the direction of v:

v=1/abs(v)⋅v

Example structure:

1/abs(v)*((v_1),(v_2),(v_3))

Distance Between Two Vectors

Distance between u and v:

dist(u,v)=abs(u-v)

Expanded: dist(u,v)=√(,((u_1)-(v_1))2+((u_2)-(v_2))2+((u_3)-(v_3))2)

Orthogonality

Vectors u and v are orthogonal if:

u⋅v=0

Orthogonal ⇔ Perpendicular in ℝn

Pythagorean Theorem (Vector Version)
If u⋅v=0

abs2(u+v)=abs2(u)+abs2(v)

Parallelogram Law

For any vectors u*v:

abs2(u+v) +abs2(u-v)=2*abs(u2)+2*abs(v2)

Finding a Unit Vector

Steps:

1. Compute abs(v)

2. Multiply vector by 1/abs(v)

Checking Orthogonality

Compute dot product

If result=0 then it is orthogonal

Distance between vectors

Subtract → take norm