Length, Distance, and Orthogonality Flashcards
the length of a vector
AKA norm or magnitude is the non-negative scalar ||v||
||v|| = √v • v = √v₁² + v₂² + … + vₐ ≥ 0
normalization of v
( 1 / ||v|| ) v
a unit vector in the same direction of v, if v is not a zero vector
distance between u and v
d(u,v) = || u - v || = √(u₁ - v₁) ² + (u₂ - v₂)² + … + (uₓ - vₓ)²
the angle between u and v
if both are nonzero vectors, cosθ = (u ⋅ v) / ( ||u|| ||v|| )
where 0 ≤ θ ≤ π
orthogonal
perpendicular, including zero vectors which are considered orthogonal to all vectors
the orthogonal projection of u onto v
projᵥ U = ( (u ⋅ v) / (v ⋅ v) ) v = ( (u ⋅ v) / ||v||² ) v if v ≠ 0
this is parallel to v
what is the vector from the tail of projᵥ U to the tail of u
u - projᵥ U
it creates a triangle where projᵥ U is orthogonal to u - projᵥ U and u is the hypotenuse