Vectors Flashcards
What does “a + b = b + a” show about vectors?
They are commutative
What does “a + (b + c) = (a + b) + c ≡ a + b + c” show about vectors?
Addition is associative
What does “λ(a + b) = λa + λb” show about vectors?
They are distributive over addition
When would velocity and acceleration not be parallel?
Orbits
Use Newton’s 2nd law to work out position of an object under constant force
F = ma a = F/m v = tF/m + v_0 s = (t^2)F/(2m) + v_0 t + r_0
Define scalar product
a · b ≡ |a||b| cos θ
What can be understood from a · b = 0?
a and b are perpendicular
OR
a or b is zero vector
Resolve vector a into parallel and perpendicular components when at angle θ to unit vector n̂
parallel: (a · n̂)n̂
perpendicular: a - (a · n̂)n̂
What shows scalar product is commutative?
a · b = b · a
How else can “λ(a · b)” be rewritten?
(λa) · b
a · (λb)
What shows scalar product is distributive over addition?
a · (b + c) = a · b + a · c
Define vector product
a x b
Vector of magnitude |a||b|sin θ
Direction perpendicular to both a and b (normal to plane of a and b)
Why would a x b = 0?
a and b are parallel
OR
a or b is zero vector
What is a x a equal to?
0
Why is a · (a × b) equal to 0?
a x b is perpendicular to a
What is |a × b| equal to (geometric)?
Area of the parallelogram formed by a and b
Vector product is anti-commutative, meaning?
a x b = -b x a
Vector product is distributive over addition, meaning?
a x (b + c) = a x b + a x c
Vector product is non-associative, meaning?
a x (b x c) =/= (a x b) x c
What is vector area equal to?
Sn̂
What is an easier way to work out the vector area of a complicated surface?
Close it as it will be equal to 0
Scalar triple product
[a,b,c] = a (b x c)
What does it mean for scalar triple product to be invariant under cyclic permutations?
[a,b,c] = [c,a,b] = [b,c,a]
Vector triple product, what would a x (b x c) equal?
a x (b x c) = (ac)b - (ab)c
Straight line general equation
r = a + λl̂
Plane general equation
r = a + λl + μq
Alternative form to general plane equation
r.n̂ = d
Distance of point B (vector b) from line r = a + λl̂
d = |l̂ x (b-a)|
Orthogonal basis
r = λa + μb + νc
How are values for components in the orthogonal basis found?
Reciprocal basis
A = (b x c)/[a,b,c]
A.r = λA.a = λ
Scalar product in component form for a.b
a.b = a(x)b(x) + a(y)b(y) + a(z)b(z)
Vector product in component form a x b
a x b = (a(y)b(z) - a(z)b(y))i + (a(z)b(x) - a(x)b(z))j + (a(x)b(y) - a(y)b(x))k
How can vector product be written as?
Determinant of a 3 by 3 matrix
How are plane polar and Cartesian coordinates related?
x = rcos(θ) y = rsin(θ) r = sqrt(x^2 + y^2) Φ = arctan(y/x)
Area element for plane polar
r dr dΦ
How are cylindrical polar and Cartesian coordinates related?
x = rcos(Φ) y = rsin(Φ) z = z 0 =< r =< infinity 0 =< Φ < 2π -infinity < z < infinity
Volume element for cylindrical polar
r dr dΦ dz
How are spherical polar and Cartesian coordinates related?
x = rsin(θ)cos(Φ) y = rsin(θ)sin(Φ) z = rcos(θ) r >= 0 0 =< θ =< π 0 =< Φ =< 2π
Volume element for spherical polar
r^2 sin(θ) dr dθ dΦ
Distance of point A, vector a, from plane r.n̂ = l
distance = |l - (a.n̂)|
