Magnitude: 𝑭 = 𝑭' + 𝑭(
!|𝒂|! = %(𝑎! " ) + *𝑎# " + + (𝑎$ " ) !|𝑭|! = %*𝐹)! " + + T𝐹)" " U + *𝐹$ " + 𝑖𝑛 𝑁𝑒𝑤𝑡𝑜𝑛𝑠
𝑭
Y=
𝑭 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Direction of vector (2D): ||𝑭||
𝑎! = !|𝒂|! cos 𝜃
𝑎# = !|𝒂|! sin 𝜃 Cross product:
𝑎" 𝑏& − 𝑎& 𝑏"
Dot Product: 𝒄 = 𝒂 × 𝒃 = \𝑎& 𝑏% − 𝑎% 𝑏& ]
𝒂 ⋅ 𝒃 = 𝑎% 𝑏% + 𝑎" 𝑏" + 𝑎& 𝑏& 𝑎% 𝑏" − 𝑎" 𝑏%
Angles between vectors: Orthogonality of CP:
𝑙𝑒𝑡 𝑎𝑛𝑔𝑙𝑒 𝑏𝑡𝑤𝑛 𝒂 𝑎𝑛𝑑 𝒃 𝑏𝑒 𝜃: 𝒄 = 𝒂 × 𝒃 𝑖𝑠 ⊥ 𝑡𝑜 𝒂 𝑎𝑛𝑑 𝒃 𝑜𝑟 𝒄 𝑖𝑠
" ⊥ 𝑡𝑜 𝑝𝑙𝑎𝑛𝑒 𝑜𝑓 𝒂 𝑎𝑛𝑑 𝒃
!|𝒂 − 𝒃|! = 𝑎" + 𝑏 " − 2𝑎𝑏 cos 𝜃 𝒂∙𝒄=0
(𝑎! − 𝑏! )" + (𝑎# − 𝑏# )" + (𝑎$ − 𝑏$ )" 𝒃∙𝒄=0
= 𝑎! " + 𝑎# " + 𝑎$ " + 𝑏! " + 𝑏# "
+ 𝑏$ " − 2𝑎𝑏 cos 𝜃 Length of 𝒂 × 𝒃: 𝐶𝑃 𝑓𝑜𝑟 2 𝑐𝑜𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 =
−2*𝑎! 𝑏! + 𝑎# 𝑏# + 𝑎$ 𝑏$ + = −2𝑎𝑏 cos 𝜃 0 𝑣𝑒𝑐𝑡𝑜𝑟 (∠ 𝑏𝑡𝑤𝑛 𝒂 𝑎𝑛𝑑 𝒃 = 0°)
𝑎! 𝑏! + 𝑎# 𝑏# + 𝑎$ 𝑏$ = 𝑎𝑏 cos 𝜃 = 𝒂 ⋅ 𝒃 !|𝒄|! = !|𝒂 × 𝒃|! = !|𝒂|!. !|𝒃|!. sin 𝜃 →
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
Orthogonal:
𝒂 ⋅ 𝒃 = 𝑎% 𝑏% + 𝑎" 𝑏" + 𝑎& 𝑏& = 0 ∴ 90° 𝒂 × 𝒃 = −𝒃 × 𝒂
𝒂 × (𝜆𝒃) = (𝜆𝒂) × 𝒃
Orthogonal set: 𝒂 × (𝒃 + 𝒄) = 𝒂 × 𝒃 + 𝒂 × 𝒄
𝒂∙𝒃=0 𝒂 × (𝒃 × 𝒄) = (𝒂 ⋅ 𝒄)𝒃 − (𝒂 ⋅ 𝒃)𝒄
𝒃∙𝒄=0
𝒄∙𝒂=0 LINES: Vector Equation: of line through (A,B)
𝒂 = 𝒓( − 𝒓'
Length of vector: → 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟𝑦 𝑏𝑡𝑤𝑛 2 𝑘𝑛𝑜𝑤𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑛 𝑙𝑖𝑛𝑒
!|𝒂|! = √𝒂 ⋅ 𝒂 𝒂 𝑎𝑛𝑑 𝒓(
− 𝒓' 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒 𝑏𝑦 𝑠𝑐𝑎𝑙𝑎𝑟
𝒓 = 𝒓( + 𝑡𝒂
Position vectors:
𝒓 = 𝒓' + 𝑡(−𝒂)
𝒓' = 𝑥' 𝒊 + 𝑦' 𝒋
𝒓 = 𝒓' + 𝑡*𝑘(𝒂)+, 𝑘 𝑖𝑠 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑠𝑐𝑎𝑙𝑎𝑟
𝒓( = 𝑥( 𝒊 + 𝑦( 𝒋
𝒓( = 𝒓' − 𝒓'( LINES: Parametric Equations:
𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌 = (𝑥( + 𝑡𝑎! )𝒊 + *𝑦( + 𝑡𝑎# +𝒋
𝒓'( = 𝒓( − 𝒓' = (𝑥( 𝒊 + 𝑦( 𝒋) − (𝑥' 𝒊 + 𝑦' 𝒋) + (𝑧( + 𝑡𝑎$ )𝒌
= (𝑥( − 𝑥' )𝒊 − (𝑦( + 𝑦' )𝒋 𝑥 = 𝑥( + 𝑎! 𝑡
𝑦 = 𝑦( + 𝑎# 𝑡
Unit vector: 𝑧 = 𝑧( + 𝑎$ 𝑡
𝒂
L=
𝒂
||𝒂|| LINES: Symmetric Equations:
𝑥 − 𝑥( 𝑦 − 𝑦( 𝑧 − 𝑧(
𝑡= = =
L
Direction cosines (3 components of unit vector 𝒂 𝑎! 𝑎# 𝑎$
– angles btwn 2 edges):
𝑛! = cos 𝛼 Vector Equation of plane:
𝑛# = cos 𝛽 𝒏 ⋅ (𝒓 − 𝒓' ) = 𝟎 ∴ ⊥
𝑛$ = cos 𝛾
Cartesian Equation:
Forces (3D): 𝑎(𝑥 − 𝑥' ) + 𝑏(𝑦 − 𝑦' ) + 𝑐(𝑧 − 𝑦$ ) = 0
𝑭' = 𝑥𝑁 × 𝒂 L ⟹ 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0
!|𝒂|! = %(𝑎! " ) + *𝑎# " + + (𝑎$ " ) !|𝑭|! = %*𝐹)! " + + T𝐹)" " U + *𝐹$ " + 𝑖𝑛 𝑁𝑒𝑤𝑡𝑜𝑛𝑠
𝑭
Y=
𝑭 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Direction of vector (2D): ||𝑭||
𝑎! = !|𝒂|! cos 𝜃
𝑎# = !|𝒂|! sin 𝜃 Cross product:
𝑎" 𝑏& − 𝑎& 𝑏"
Dot Product: 𝒄 = 𝒂 × 𝒃 = \𝑎& 𝑏% − 𝑎% 𝑏& ]
𝒂 ⋅ 𝒃 = 𝑎% 𝑏% + 𝑎" 𝑏" + 𝑎& 𝑏& 𝑎% 𝑏" − 𝑎" 𝑏%
Angles between vectors: Orthogonality of CP:
𝑙𝑒𝑡 𝑎𝑛𝑔𝑙𝑒 𝑏𝑡𝑤𝑛 𝒂 𝑎𝑛𝑑 𝒃 𝑏𝑒 𝜃: 𝒄 = 𝒂 × 𝒃 𝑖𝑠 ⊥ 𝑡𝑜 𝒂 𝑎𝑛𝑑 𝒃 𝑜𝑟 𝒄 𝑖𝑠
" ⊥ 𝑡𝑜 𝑝𝑙𝑎𝑛𝑒 𝑜𝑓 𝒂 𝑎𝑛𝑑 𝒃
!|𝒂 − 𝒃|! = 𝑎" + 𝑏 " − 2𝑎𝑏 cos 𝜃 𝒂∙𝒄=0
(𝑎! − 𝑏! )" + (𝑎# − 𝑏# )" + (𝑎$ − 𝑏$ )" 𝒃∙𝒄=0
= 𝑎! " + 𝑎# " + 𝑎$ " + 𝑏! " + 𝑏# "
+ 𝑏$ " − 2𝑎𝑏 cos 𝜃 Length of 𝒂 × 𝒃: 𝐶𝑃 𝑓𝑜𝑟 2 𝑐𝑜𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 =
−2*𝑎! 𝑏! + 𝑎# 𝑏# + 𝑎$ 𝑏$ + = −2𝑎𝑏 cos 𝜃 0 𝑣𝑒𝑐𝑡𝑜𝑟 (∠ 𝑏𝑡𝑤𝑛 𝒂 𝑎𝑛𝑑 𝒃 = 0°)
𝑎! 𝑏! + 𝑎# 𝑏# + 𝑎$ 𝑏$ = 𝑎𝑏 cos 𝜃 = 𝒂 ⋅ 𝒃 !|𝒄|! = !|𝒂 × 𝒃|! = !|𝒂|!. !|𝒃|!. sin 𝜃 →
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
Orthogonal:
𝒂 ⋅ 𝒃 = 𝑎% 𝑏% + 𝑎" 𝑏" + 𝑎& 𝑏& = 0 ∴ 90° 𝒂 × 𝒃 = −𝒃 × 𝒂
𝒂 × (𝜆𝒃) = (𝜆𝒂) × 𝒃
Orthogonal set: 𝒂 × (𝒃 + 𝒄) = 𝒂 × 𝒃 + 𝒂 × 𝒄
𝒂∙𝒃=0 𝒂 × (𝒃 × 𝒄) = (𝒂 ⋅ 𝒄)𝒃 − (𝒂 ⋅ 𝒃)𝒄
𝒃∙𝒄=0
𝒄∙𝒂=0 LINES: Vector Equation: of line through (A,B)
𝒂 = 𝒓( − 𝒓'
Length of vector: → 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟𝑦 𝑏𝑡𝑤𝑛 2 𝑘𝑛𝑜𝑤𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑛 𝑙𝑖𝑛𝑒
!|𝒂|! = √𝒂 ⋅ 𝒂 𝒂 𝑎𝑛𝑑 𝒓(
− 𝒓' 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒 𝑏𝑦 𝑠𝑐𝑎𝑙𝑎𝑟
𝒓 = 𝒓( + 𝑡𝒂
Position vectors:
𝒓 = 𝒓' + 𝑡(−𝒂)
𝒓' = 𝑥' 𝒊 + 𝑦' 𝒋
𝒓 = 𝒓' + 𝑡*𝑘(𝒂)+, 𝑘 𝑖𝑠 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑠𝑐𝑎𝑙𝑎𝑟
𝒓( = 𝑥( 𝒊 + 𝑦( 𝒋
𝒓( = 𝒓' − 𝒓'( LINES: Parametric Equations:
𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌 = (𝑥( + 𝑡𝑎! )𝒊 + *𝑦( + 𝑡𝑎# +𝒋
𝒓'( = 𝒓( − 𝒓' = (𝑥( 𝒊 + 𝑦( 𝒋) − (𝑥' 𝒊 + 𝑦' 𝒋) + (𝑧( + 𝑡𝑎$ )𝒌
= (𝑥( − 𝑥' )𝒊 − (𝑦( + 𝑦' )𝒋 𝑥 = 𝑥( + 𝑎! 𝑡
𝑦 = 𝑦( + 𝑎# 𝑡
Unit vector: 𝑧 = 𝑧( + 𝑎$ 𝑡
𝒂
L=
𝒂
||𝒂|| LINES: Symmetric Equations:
𝑥 − 𝑥( 𝑦 − 𝑦( 𝑧 − 𝑧(
𝑡= = =
L
Direction cosines (3 components of unit vector 𝒂 𝑎! 𝑎# 𝑎$
– angles btwn 2 edges):
𝑛! = cos 𝛼 Vector Equation of plane:
𝑛# = cos 𝛽 𝒏 ⋅ (𝒓 − 𝒓' ) = 𝟎 ∴ ⊥
𝑛$ = cos 𝛾
Cartesian Equation:
Forces (3D): 𝑎(𝑥 − 𝑥' ) + 𝑏(𝑦 − 𝑦' ) + 𝑐(𝑧 − 𝑦$ ) = 0
𝑭' = 𝑥𝑁 × 𝒂 L ⟹ 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0
