Length, Distance, and Orthogonality

Question 1

the length of a vector

Answer 1

AKA norm or magnitude is the non-negative scalar ||v||

||v|| = √v • v = √v₁² + v₂² + … + vₐ ≥ 0

Question 2

normalization of v

Answer 2

( 1 / ||v|| ) v

a unit vector in the same direction of v, if v is not a zero vector

Question 3

distance between u and v

Answer 3

d(u,v) = || u - v || = √(u₁ - v₁) ² + (u₂ - v₂)² + … + (uₓ - vₓ)²

Question 4

the angle between u and v

Answer 4

if both are nonzero vectors, cosθ = (u ⋅ v) / ( ||u|| ||v|| )
where 0 ≤ θ ≤ π

Question 5

orthogonal

Answer 5

perpendicular, including zero vectors which are considered orthogonal to all vectors

Question 6

the orthogonal projection of u onto v

Answer 6

projᵥ U = ( (u ⋅ v) / (v ⋅ v) ) v = ( (u ⋅ v) / ||v||² ) v if v ≠ 0
this is parallel to v

Question 7

what is the vector from the tail of projᵥ U to the tail of u

Answer 7

u - projᵥ U

it creates a triangle where projᵥ U is orthogonal to u - projᵥ U and u is the hypotenuse