Vectors cheat sheet - a summary to vector calculus PDF

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Vectors Cheat Sheet Vecto Vectorr Magnit gnitude ude ‖〈𝑎, 𝑏 〉‖ = √𝑎2 + 𝑏 2

Unit Vecto ctorr (No Norma rma rmalizing lizing a V Vecto ecto ector) r)

Add Addin in ing g Vect ector or orss

Scalar M Multipli ultipli ultiplication cation

(𝑎𝚤 + 𝑏𝚥) + (𝑐𝚤 + 𝑑𝚥) = (𝑎 + 𝑐)𝚤 + (𝑏 + 𝑑)𝚥

𝑎(𝑏𝚤 + 𝑐𝚥) = 𝑎𝑏𝚤 + 𝑎𝑐𝚥

Divide by the magnitude:

〈𝑎, 𝑏〉 + 〈𝑐, 𝑑〉 = 〈𝑎 + 𝑐, 𝑏 + 𝑑 〉

𝑎〈𝑏, 𝑐〉 = 〈𝑎𝑏, 𝑎𝑐〉

Do Dott P Prod rod roducts ucts

〈𝑎, 𝑏〉 ∙ 〈𝑐, 𝑑〉 = 𝑎𝑐 + 𝑏𝑑

(𝑎𝚤 + 𝑏𝚥) × (𝑐𝚤 + 𝑑𝚥) = 𝑎𝑐 + 𝑏𝑑

Alternative formula: 𝑢� ∙ 𝑣 = ‖�𝑢 ‖ ‖𝑣‖ cos 𝜃

Cross Products

〈𝑎, 𝑏 , 𝑐〉 × 〈𝑑, 𝑒, 𝑓〉 = 〈𝑏𝑓 − 𝑒𝑐, − (𝑎𝑓 − 𝑑𝑐), 𝑎𝑒 − 𝑑𝑏〉 𝚤 �𝑎 𝑑

𝚥 𝑏 𝑒

𝑘� 𝑐 � = (𝑏𝑓 − 𝑒𝑐 )𝚤 − (𝑎𝑓 − 𝑑𝑐)𝚥 + (𝑎𝑒 − 𝑑𝑏 )𝑘� 𝑓

Magnitude of the cross product: ‖𝑢� × 𝑣‖ = ‖�𝑢 ‖ ‖𝑣‖ sin 𝜃

Projections 𝑝𝑟𝑜𝑗𝑢 𝑣 =

� 𝑢� ∙𝑣 𝑢� ‖𝑢� ‖2

Vector Angles

𝑝𝑟𝑜𝑗𝑣�𝑢 =

� 𝑢� ∙𝑣 𝑣 ‖𝑣�‖2

Finding the direction of a single vector: tan 𝜃 =

𝑦 𝑥

Finding the angle separating two vectors: cos 𝜃 =

=

𝚥 𝑡𝑒𝑟𝑚 𝚤 𝑡𝑒𝑟𝑚

𝑢� ∙𝑣� ‖𝑢 �� ‖ ‖𝑣� ‖

𝑢�� ‖𝑢�� ‖...