CSCI 8945 · In-class tool
Inner, outer, and cross products of 2D vectors
Drag the sliders to change vectors u and v. All three panels update live, so you can see how the same two vectors produce three very different kinds of results — a scalar length, a transformation, and a signed area with direction.
Vector u
Vector v
Inner product
u·v = u₁v₁ + u₂v₂ = |u||v|cosθ
shown as the projection of v onto u.
u·v = —
length of v projected onto u = —
length of v projected onto u = —
Outer product
u⊗v = uvT, a rank-1 matrix. Applied to a vector w it gives (u⊗v)w = u(v·w) — always a multiple of u.
u⊗v =
——
——
v·w = —
(u⊗v)w = —
(u⊗v)w = —
Third vector w (for the transformation)
Cross product
u×v = u₁v₂ − u₂v₁. Shown here in 3D: u and v lie flat in the xy-plane, and the normal n = u×v rises straight up or dips straight down out of that plane, with length equal to the parallelogram's area.
Drag to rotate · double-click to reset the view
u×v = —
parallelogram area = —, normal n —
parallelogram area = —, normal n —