Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Classical Mechanics (Kibble & Berkshire) · Problem 1.12
Velocity, acceleration and scalar angular velocity — T.W.B. Kibble, F.H. Berkshire
Physics — Mechanics · vector-algebra-reciprocal-basis
Problem Reciprocal basis Let { a , b , c } \{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\} { a , b , c } { a ∗ , b ∗ , c ∗ } \{\boldsymbol{a}^*, \boldsymbol{b}^*, \boldsymbol{c}^*\} { a ∗ , b ∗ , c ∗ } a ∗ = b × c [ a , b , c ] , b ∗ = c × a [ a , b , c ] , c ∗ = a × b [ a , b , c ] . \boldsymbol{a}^* = \frac{\boldsymbol{b}\times\boldsymbol{c}}{[\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}]}, \quad \boldsymbol{b}^* = \frac{\boldsymbol{c}\times\boldsymbol{a}}{[\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}]}, \quad \boldsymbol{c}^* = \frac{\boldsymbol{a}\times\boldsymbol{b}}{[\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}]}. a ∗ = [ a , b , c ] b × c , b ∗ = [ a , b , c ] c × a , c ∗ = [ a , b , c ] a × b . { i , j , k } \{\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\} { i , j , k } { i ∗ , j ∗ , k ∗ } = { i , j , k } \{\boldsymbol{i}^*, \boldsymbol{j}^*, \boldsymbol{k}^*\} = \{\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\} { i ∗ , j ∗ , k ∗ } = { i , j , k } [ a ∗ , b ∗ , c ∗ ] = 1 / [ a , b , c ] [\boldsymbol{a}^*, \boldsymbol{b}^*, \boldsymbol{c}^*] = 1/[\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}] [ a ∗ , b ∗ , c ∗ ] = 1/ [ a , b , c ] { a , b , c } \{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\} { a , b , c } { a ∗ , b ∗ , c ∗ } \{\boldsymbol{a}^*, \boldsymbol{b}^*, \boldsymbol{c}^*\} { a ∗ , b ∗ , c ∗ } { ( a ∗ ) ∗ , ( b ∗ ) ∗ , ( c ∗ ) ∗ } = { a , b , c } \{(\boldsymbol{a}^*)^*, (\boldsymbol{b}^*)^*, (\boldsymbol{c}^*)^*\} = \{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\} {( a ∗ ) ∗ , ( b ∗ ) ∗ , ( c ∗ ) ∗ } = { a , b , c } v \boldsymbol{v} v { a , b , c } \{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\} { a , b , c } v = λ a + μ b + ν c , \boldsymbol{v} = \lambda \boldsymbol{a} + \mu \boldsymbol{b} + \nu \boldsymbol{c}, v = λ a + μ b + ν c , λ \lambda λ μ \mu μ ν \nu ν λ = v ⋅ a ∗ \lambda = \boldsymbol{v} \cdot \boldsymbol{a}^* λ = v ⋅ a ∗ μ = v ⋅ b ∗ \mu = \boldsymbol{v} \cdot \boldsymbol{b}^* μ = v ⋅ b ∗ ν = v ⋅ c ∗ \nu = \boldsymbol{v} \cdot \boldsymbol{c}^* ν = v ⋅ c ∗ Browse the problem library