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Vectors - Coggle Diagram
Vectors
Equations of lines
Vector form
r = a + λb
Cartesian form
x - a(1) = y - a(2) = z - a(3)
-̅-̅b̅(̅1̅)̅------̅-̅-̅b̅(̅2̅)̅-̅------̅-̅b̅(̅3̅)̅-̅
Vector product form
(r - a) x b = 0
Equation of planes
Vector from
r = a + λb + μc
Cartesian form
ax + by + cz = p
Scalar form
r ⋅ n = a ⋅ n = p
Vector product
a x b = |a| |b| sinθ n̂ = | i j k |
| a a a |
| b b b |
Parallel
a x b = 0
Area of triangle
1/2 |a x b|
Other
sinθ = |n ⋅ b|
|̅n|̅ ̅|̅b̅|̅
Perpendicular distance between point and plane
|a ⋅ n - p|
̅ ̅ ̅ ̅ ̅|̅n|̅ ̅ ̅ ̅ ̅
Distance between skew lines
|(a(1) - a(2)) ⋅ (b(1) x b(2))|
̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅|̅b(1) ̅x ̅b(2)|̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
Scalar product
a ⋅ b = |a| |b| cosθ = a(1)b(1) + a(2)b(2) + a(3)b(3)
Perpendicular
a ⋅ b = 0