Length, Distance, and Orthogonality Flashcards

1
Q

the length of a vector

A

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

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

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2
Q

normalization of v

A

( 1 / ||v|| ) v

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

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3
Q

distance between u and v

A

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

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4
Q

the angle between u and v

A

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

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5
Q

orthogonal

A

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

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6
Q

the orthogonal projection of u onto v

A

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

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7
Q

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

A

u - projᵥ U

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

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