Vectors

Size and direction

Magnitude

‖ 𝐯 ‖ = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}

Unit vector

𝐮 = \frac{𝐯}{‖ 𝐯 ‖}

Dot product

𝐮 \cdot 𝐯 = u_{x} v_{x} + u_{y} v_{y} + u_{z} v_{z} = ‖ 𝐮 ‖ ‖ 𝐯 ‖ \cos θ

u · v = 0 means perpendicular

Projection of u onto v

{proj}_{𝐯} 𝐮 = \frac{𝐮 \cdot 𝐯}{‖ 𝐯 ‖^{2}} 𝐯

Cross product

𝐮 \times 𝐯 = ⟨ u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1} ⟩

Size of the cross product

‖ 𝐮 \times 𝐯 ‖ = ‖ 𝐮 ‖ ‖ 𝐯 ‖ \sin θ

area of the parallelogram

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