vectors cheat sheet, Cheat Sheet of Mathematics

Download vectors cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! Vectors Cheat Sheet Vector Magnitude ‖〈𝑎, 𝑏〉‖ = √𝑎2 + 𝑏2 Unit Vector (Normalizing a Vector) Divide by the magnitude: 𝑢�⃑‖𝑢�⃑ ‖ Adding Vectors 〈𝑎, 𝑏〉 + 〈𝑐,𝑑〉 = 〈𝑎 + 𝑐, 𝑏 + 𝑑〉 (𝑎𝚤̂ + 𝑏𝚥̂) + (𝑐𝚤̂ + 𝑑𝚥̂) = (𝑎 + 𝑐)𝚤̂ + (𝑏 + 𝑑)𝚥̂ Scalar Multiplication 𝑎〈𝑏, 𝑐〉 = 〈𝑎𝑏,𝑎𝑐〉 𝑎(𝑏𝚤̂ + 𝑐𝚥̂) = 𝑎𝑏𝚤̂ + 𝑎𝑐𝚥̂ Dot Products 〈𝑎, 𝑏〉 ∙ 〈𝑐,𝑑〉 = 𝑎𝑐 + 𝑏𝑑 (𝑎𝚤̂ + 𝑏𝚥̂) × (𝑐𝚤̂ + 𝑑𝚥̂) = 𝑎𝑐 + 𝑏𝑑 Alternative formula: 𝑢�⃑ ∙ ?⃑? = ‖𝑢�⃑ ‖ ‖?⃑?‖ cos𝜃 Cross Products 〈𝑎, 𝑏, 𝑐〉 × 〈𝑑, 𝑒,𝑓〉 = 〈𝑏𝑓 − 𝑒𝑐, − (𝑎𝑓 − 𝑑𝑐), 𝑎𝑒 − 𝑑𝑏〉 � 𝚤̂ 𝚥̂ 𝑘� 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 � = (𝑏𝑓 − 𝑒𝑐)𝚤̂ − (𝑎𝑓 − 𝑑𝑐)𝚥̂ + (𝑎𝑒 − 𝑑𝑏)𝑘� Magnitude of the cross product: ‖𝑢�⃑ × ?⃑?‖ = ‖𝑢�⃑ ‖ ‖?⃑?‖ sin𝜃 Projections 𝑝𝑟𝑜𝑗𝑢?⃑? = 𝑢�⃑ ∙𝑣�⃑ ‖𝑢�⃑ ‖2 𝑢�⃑ 𝑝𝑟𝑜𝑗𝑣𝑢�⃑ = 𝑢�⃑ ∙𝑣�⃑ ‖𝑣�⃑ ‖2 ?⃑? Vector Angles Finding the direction of a single vector: tan𝜃 = 𝑦 𝑥 = ?̂? 𝑡𝑒𝑟𝑚 ?̂? 𝑡𝑒𝑟𝑚 Finding the angle separating two vectors: cos𝜃 = 𝑢�⃑ ∙𝑣�⃑‖𝑢�⃑ ‖ ‖𝑣�⃑ ‖