Vectors - Formula Sheet

Vectors – Formula Sheet: 2D Vectors: 𝑽𝑿 = 𝑉𝑐𝑜𝑠(𝜃) 𝑽𝒀 = 𝑉𝑠𝑖𝑛(𝜃) 𝜽 = 𝑡𝑎𝑛−1 ( 𝑉𝑌 ) 𝑉𝑋 |𝑽| = √𝑉𝑋2 + 𝑉𝑌2 ⃗ = |𝑉 |𝑐𝑜𝑠(𝜃)𝒊 + |𝑉|𝑠𝑖𝑛(𝜃)𝒋 𝑽 Scalar Dot Product: ⃗𝑨 ⃗ ∙𝑩 ⃗⃗ = |𝐴||𝐵 |𝑐𝑜𝑠(𝜃) = 𝐴𝑋 𝐵𝑋 + 𝐴𝑌 𝐵𝑌 + 𝐴𝑍 𝐵𝑍 |𝑨| = √𝐴𝑋2 + 𝐴2𝑌 + 𝐴2𝑍 |𝑩| = √𝐵𝑋2 + 𝐵𝑌2 + 𝐵𝑍2 Vector Cross Product: 𝒊 ⃗⃗ 𝒙𝑩 ⃗⃗ = |𝐴| |𝐵|𝑠𝑖𝑛(𝜃)𝒓̂ = |𝐴𝑋 𝑨 𝐵𝑋 𝒋 𝐴𝑌 𝐵𝑌 𝒌 𝐴𝑍 | 𝐵𝑍 ⃗⃗ 𝒙𝑩 ⃗⃗ = (𝐴𝑌 𝐵𝑍 − 𝐴𝑍 𝐵𝑌 )𝒊 + (𝐴𝑍 𝐵𝑋 − 𝐴𝑋 𝐵𝑍 )𝒋 + (𝐴𝑋 𝐵𝑌 − 𝐴𝑌 𝐵𝑋 )𝒌 𝑨 Unit Vector – A vector with a magnitude of 1 that is used to give direction to another vector. ⃗⃗ = 𝑼 ⃗ 𝑉 |𝑉| Standard Unit Vectors: 𝑖 = < 1, 0, 0 > 𝑗 = < 0,1,0 > 𝑘 = < 0,0,1 > 𝑖𝑥𝑗=𝑘 𝑗𝑥𝑘=𝑖 𝑘𝑥𝑖=𝑗 𝑖∙𝑖 =1 𝑗∙𝑗 =1 𝑘∙𝑘 =1 𝑗 𝑥 𝑖 = −𝑘 𝑘 𝑥 𝑗 = −𝑖 𝑖 𝑥 𝑘 = −𝑗 𝑖∙𝑗=0 𝑗∙𝑘 =0 𝑖∙𝑘 =0 𝑖𝑥𝑖=0 𝑗𝑥𝑗=0 𝑘𝑥𝑘=0 Position Vector: ⃗⃗⃗⃗⃗⃗⃗ 𝒓𝑨𝑩 = (𝑋𝐵 − 𝑋𝐴 )𝒊 + (𝑌𝐵 − 𝑌𝐴 )𝒋 + (𝑍𝐵 − 𝑍𝐴 )𝒌 ⃗⃗⃗⃗⃗⃗⃗ 𝒓𝑨𝑩 = 𝑟𝑥 𝒊 + 𝑟𝑦 𝒋 + 𝑟𝑧 𝒌 𝑟𝑥 = 𝑋𝐵 − 𝑋𝐴 𝑟𝑦 = 𝑌𝐵 − 𝑌𝐴 𝑟𝑧 = 𝑍𝐵 − 𝑍𝐴 2 2 2 |𝒓 ⃗⃗⃗⃗⃗⃗⃗ 𝑨𝑩 | = √𝑟𝑥 + 𝑟𝑦 + 𝑟𝑧 𝒓̂ = 𝑟𝑥 𝒊 + 𝑟𝑦 𝒋 + 𝑟𝑧 𝒌 𝑟⃗⃗⃗⃗⃗ 𝐴𝐵 = |𝑟| √𝑟𝑥2 + 𝑟𝑦2 + 𝑟𝑧2 Vector Addition: ⃗⃗ = 𝐴 + 𝐵 ⃗ 𝑹 ⃗⃗ = (𝐴𝑋 + 𝐵𝑋 )𝒊 + (𝐴𝑌 + 𝐵𝑌 )𝒋 + (𝐴𝑍 + 𝐵𝑍 )𝒌 𝑹 |𝑹| = √|𝐴|2 + |𝐵|2 + 2|𝐴||𝐵|𝑐𝑜𝑠(𝜃) Finding The Angle Between Two Vectors: 𝜽 = 𝑐𝑜𝑠 𝜽 = 𝑐𝑜𝑠 −1 ( Scalar Projection: −1 ⃗⃗⃗ ⃗ 𝐴 ∙𝐵 ( ) |𝐴| |𝐵| 𝐴𝑋 𝐵𝑋 + 𝐴𝑌 𝐵𝑌 + 𝐴𝑍 𝐵𝑍 √𝐴𝑥2 + 𝐴2𝑌 + 𝐴2𝑍 × √𝐵𝑋2 + 𝐵𝑌2 + 𝐵𝑍2 Vector Projection: 𝑪𝒐𝒎𝒑𝒂⃗ ⃗𝒃 = 𝑎 ∙ 𝑏⃗ |𝑎| "𝒃 𝒐𝒏𝒕𝒐 𝒂" 𝑷𝒓𝒐𝒋𝒂⃗ ⃗𝒃 = 𝑎 ∙ 𝑏⃗ ∙ 𝑎 |𝑎|2 "𝒃 𝒐𝒏𝒕𝒐 𝒂" ⃗ = 𝑪𝒐𝒎𝒑𝒃⃗ 𝒂 𝑎 ∙ 𝑏⃗ |𝑏| "𝒂 𝒐𝒏𝒕𝒐 𝒃" ⃗ = 𝑷𝒓𝒐𝒋𝒃⃗ 𝒂 𝑎 ∙ 𝑏⃗ ∙ 𝑏⃗ 2 |𝑏| "𝒂 𝒐𝒏𝒕𝒐 𝒃" ) Scalar Triple Product: 𝐴𝑋 ⃗𝑨 ⃗ ∙ (𝑩 ⃗⃗ 𝒙 ⃗𝑪) = 𝑑𝑒𝑡 |𝐵𝑋 𝐶𝑋 𝐴𝑌 𝐵𝑌 𝐶𝑌 𝐴𝑍 𝐵𝑍 | 𝐶𝑍 = (𝐴𝑌 𝐵𝑍 − 𝐴𝑍 𝐵𝑌 )𝐶𝑋 + (𝐴𝑍 𝐵𝑋 − 𝐴𝑋 𝐵𝑍 )𝐶𝑌 + (𝐴𝑋 𝐵𝑌 − 𝐴𝑌 𝐵𝑋 )𝐶𝑍 ⃗𝑨 ⃗ ∙ (𝑩 ⃗⃗ 𝒙 ⃗𝑪) = 𝐵 ⃗ ∙ (𝐶 𝑥 𝐴) = 𝐶 ∙ (𝐴 𝑥 𝐵 ⃗) ⃗𝑨 ⃗ ∙ (𝑩 ⃗⃗ 𝒙 ⃗𝑪) = −𝐴 ∙ (𝐶 𝑥 𝐵 ⃗) Vector Triple Product: ⃗𝑨 ⃗ 𝒙 (𝑩 ⃗⃗ 𝒙 ⃗𝑪) = (𝐴 ∙ 𝐶 )𝐵 ⃗ − (𝐴 ∙ 𝐵 ⃗ )𝐶 Area of a Triangle: 1 ⃗⃗⃗⃗⃗ 𝑥 𝐴𝐶 ⃗⃗⃗⃗⃗ | 𝑨𝒓𝒆𝒂 = |𝐴𝐵 2 Area of a Parallelogram: ⃗⃗⃗⃗⃗ 𝑥 ⃗⃗⃗⃗⃗ 𝑨𝒓𝒆𝒂 = |𝐴𝐵 𝐴𝐷| Volume of a Parallelepiped: (Triple Scalar Product) ⃗ 𝑥 𝐶 )| 𝑽𝒐𝒍𝒖𝒎𝒆 = |𝐴 ∙ (𝐵 Direction Angles: 𝑐𝑜𝑠(𝜶) = 𝑉𝑋 |𝑉| cos(𝜷) = 𝑉𝑌 |𝑉| cos(𝜸) = 𝑉𝑍 |𝑉| ⃗ = |𝑉| cos(𝜶) 𝒊 + |𝑉| cos(𝜷) 𝒋 + |𝑉| cos(𝜸) 𝒌 𝑉 𝑐𝑜𝑠 2 (𝜶) + 𝑐𝑜𝑠 2 (𝜷) + 𝑐𝑜𝑠 2 (𝜸) = 1 𝑉̂ = 𝑈𝑉 = 𝑉𝑋 𝑉𝑌 𝑉𝑍 𝒊+ 𝒋+ 𝒌 |𝑉| |𝑉| |𝑉| 𝑉̂ = 𝑈𝑉 = cos(𝜶) 𝒊 + cos(𝜷) 𝒋 + cos(𝜸) 𝒌