University:
University of Michigan
Course:
MATH 215 | Multivariable and Vector Calculus
Academic year:

2022
Views:

255

Pages:

1
Author:

Accinkiquicpqw

Vectors Cheat Sheet Ve Vecto cto ctorr M Maagni gnitud tud tudee Un Uniit V Veecto ctorr (Norm rmali ali alizin zin zing gaV Vecto ecto ector) r) ‖〈𝑎, 𝑏〉‖ = √𝑎2 + 𝑏 2 Divide by the magnitude: Ad Addin din ding g Vec ecto to tors rs Sca Scalar lar M Multi ulti ultipli pli plicat cat catio io ion n 〈𝑎, 𝑏〉 + 〈𝑐, 𝑑〉 = 〈𝑎 + 𝑐, 𝑏 + 𝑑〉 𝑎〈𝑏, 𝑐〉 = 〈𝑎𝑏, 𝑎𝑐〉 � � 𝑢 ‖𝑢� �‖ (𝑎𝚤 + 𝑏𝚥) + (𝑐𝚤 + 𝑑𝚥) = (𝑎 + 𝑐)𝚤 + (𝑏 + 𝑑)𝚥 𝑎(𝑏𝚤 + 𝑐𝚥) = 𝑎𝑏𝚤 + 𝑎𝑐𝚥 Do Dott P Pro ro rodu du ducts cts 〈𝑎, 𝑏〉 ∙ 〈𝑐, 𝑑〉 = 𝑎𝑐 + 𝑏𝑑 (𝑎𝚤 + 𝑏𝚥) × (𝑐𝚤 + 𝑑𝚥) = 𝑎𝑐 + 𝑏𝑑 Alternative formula: 𝑢 �∙ 𝑣 = ‖� 𝑢‖ ‖𝑣 ‖ cos 𝜃 Cro Cross ss P Pro ro rodu du ducts cts 〈𝑎, 𝑏, 𝑐〉 × 〈𝑑, 𝑒, 𝑓〉 = 〈𝑏𝑓 − 𝑒𝑐, − (𝑎𝑓 − 𝑑𝑐), 𝑎𝑒 − 𝑑𝑏〉 𝚤 𝚥 𝑘� � 𝑎 𝑏 𝑐� = (𝑏𝑓 − 𝑒𝑐 )𝚤 − (𝑎𝑓 − 𝑑𝑐)𝚥 + (𝑎𝑒 − 𝑑𝑏)𝑘� 𝑑 𝑒 𝑓 Magnitude of the cross product: ‖𝑢�× 𝑣 ‖ = ‖𝑢�‖ ‖𝑣 ‖ sin 𝜃 Pro Proje je jecti cti ction on onss � � 𝑢∙𝑣 ‖ 𝑝𝑟𝑜𝑗𝑢 𝑣 = ‖𝑢� 𝑢� �2 𝑝𝑟𝑜𝑗� 𝑣 𝑢= � 𝑢�∙𝑣 𝑣 ‖𝑣� ‖2 Ve Vecto cto ctorr Angle gless 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�‖

Vectors Cheat Sheet, Magnitude, Unit Vector

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