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
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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"
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)
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
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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
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