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Analytic Geometry2D

Vector Analytic Geometry - Theory & Concepts - Analytic Geometry Vector Projection Angle

Introduction to vectors, dot product, cross product, and their applications in analytic geometry.

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Vector Projection & Angle Explorer

Vector u = ⟨u_x, u_y⟩
u_x3.0
u_y2.0
Vector v = ⟨v_x, v_y⟩
v_x4.0
v_y0.0
Dot Product (u · v):12.00
Angle (θ):33.7°
Projection Scalar:0.750
proj_v u:⟨3.00, 0.00⟩
Vector uVector vProjection Vectoruvproj_v u
Click: change u • Shift+Click: change v
Mathematical Representation
u⃗⋅v⃗=uxvx+uyvy=(3.0)(4.0)+(2.0)(0.0)=12.00\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y = (3.0)(4.0) + (2.0)(0.0) = 12.00θ=arccos⁡(u⃗⋅v⃗∥u⃗∥∥v⃗∥)=arccos⁡(12.03.6⋅4.0)=33.7∘\theta = \arccos\left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}\right) = \arccos\left(\frac{12.0}{3.6 \cdot 4.0}\right) = 33.7^\circprojv⃗u⃗=(u⃗⋅v⃗∥v⃗∥2)v⃗=(0.75)v⃗=⟨3.00,0.00⟩\text{proj}_{\vec{v}} \vec{u} = \left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2}\right)\vec{v} = (0.75) \vec{v} = \langle 3.00, 0.00 \rangle

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