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Products
Dot Product
Definition
Given 2 nonzero vectors U and V in 2D or 3D, their dot product is U⋅V=∣U∣∣V∣cosθ where θ is the angle between U and V with 0≤θ≤π
Equation
U=⟨U1,U2,U3⟩
V=⟨V1,V2,V3⟩
U⋅V=U1V1+U2V2+U3V3=∣U∣∣V∣cosθ
Key Notes
Dot product is commutative
Orthogonal Vectors
Two vectors are orthogonal if U⋅V=0. In 2D and 3D spaces, orthogonal means perpendicular (2π angle between them)
Key Notes
There are an infinite number of orthogonal vectors to any one vector when length is not specified
Projections
The orthogonal projection is a vector that goes along U with length ∣V∣cosθ
Equation
projUV=∣U∣U∣V∣cosθ
Cross Product
Definition
Given two nonzero vectors u and v in IR3, the cross product is defined as u×v=∣u∣∣u∣sinθn where 0≤θ≤π is the angle between u and v and n is the unit vector which is normal (perpendicular) to u and v whose direction is determined by the right hand rule
Equationu×v=∣u∣∣u∣sinθn
Key Notes
u and v are parallel if and only if u×v=0
If u and v are two sides of a parallelogram, then the area of the paralllelogram is ∣u×v∣=∣u∣∣v∣sinθ
v×u=−(u×v)
u×v is orthogonal to u and v which means that (u×v)⋅u=0 and (u×v)⋅v=0