Vectors

Question 1

What does “a + b = b + a” show about vectors?

Answer 1

They are commutative

Question 2

What does “a + (b + c) = (a + b) + c ≡ a + b + c” show about vectors?

Answer 2

Addition is associative

Question 3

What does “λ(a + b) = λa + λb” show about vectors?

Answer 3

They are distributive over addition

Question 4

When would velocity and acceleration not be parallel?

Answer 4

Orbits

Question 5

Use Newton’s 2nd law to work out position of an object under constant force

Answer 5

F = ma
a = F/m
v = tF/m + v_0
s = (t^2)F/(2m) + v_0 t + r_0

Question 6

Define scalar product

Answer 6

a · b ≡ |a||b| cos θ

Question 7

What can be understood from a · b = 0?

Answer 7

a and b are perpendicular
OR
a or b is zero vector

Question 8

Resolve vector a into parallel and perpendicular components when at angle θ to unit vector n̂

Answer 8

parallel: (a · n̂)n̂
perpendicular: a - (a · n̂)n̂

Question 9

What shows scalar product is commutative?

Answer 9

a · b = b · a

Question 10

How else can “λ(a · b)” be rewritten?

Answer 10

(λa) · b

a · (λb)

Question 11

What shows scalar product is distributive over addition?

Answer 11

a · (b + c) = a · b + a · c

Question 12

Define vector product

Answer 12

a x b
Vector of magnitude |a||b|sin θ
Direction perpendicular to both a and b (normal to plane of a and b)

Question 13

Why would a x b = 0?

Answer 13

a and b are parallel
OR
a or b is zero vector

Question 14

What is a x a equal to?

Answer 14

0

Question 15

Why is a · (a × b) equal to 0?

Answer 15

a x b is perpendicular to a

Question 16

What is |a × b| equal to (geometric)?

Answer 16

Area of the parallelogram formed by a and b

Question 17

Vector product is anti-commutative, meaning?

Answer 17

a x b = -b x a

Question 18

Vector product is distributive over addition, meaning?

Answer 18

a x (b + c) = a x b + a x c

Question 19

Vector product is non-associative, meaning?

Answer 19

a x (b x c) =/= (a x b) x c

Question 20

What is vector area equal to?

Answer 20

Sn̂

Question 21

What is an easier way to work out the vector area of a complicated surface?

Answer 21

Close it as it will be equal to 0

Question 22

Scalar triple product

Answer 22

[a,b,c] = a (b x c)

Question 23

What does it mean for scalar triple product to be invariant under cyclic permutations?

Answer 23

[a,b,c] = [c,a,b] = [b,c,a]

Question 24

Vector triple product, what would a x (b x c) equal?

Answer 24

a x (b x c) = (ac)b - (ab)c

Question 25

Straight line general equation

Answer 25

r = a + λl̂

Question 26

Plane general equation

Answer 26

r = a + λl + μq

Question 27

Alternative form to general plane equation

Answer 27

r.n̂ = d

Question 28

Distance of point B (vector b) from line r = a + λl̂

Answer 28

d = |l̂ x (b-a)|

Question 29

Orthogonal basis

Answer 29

r = λa + μb + νc

Question 30

How are values for components in the orthogonal basis found?

Answer 30

Reciprocal basis
A = (b x c)/[a,b,c]
A.r = λA.a = λ

Question 31

Scalar product in component form for a.b

Answer 31

a.b = a(x)b(x) + a(y)b(y) + a(z)b(z)

Question 32

Vector product in component form a x b

Answer 32

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

Question 33

How can vector product be written as?

Answer 33

Determinant of a 3 by 3 matrix

Question 34

How are plane polar and Cartesian coordinates related?

Answer 34

x = rcos(θ)
y = rsin(θ)
r = sqrt(x^2 + y^2)
Φ = arctan(y/x)

Question 35

Area element for plane polar

Answer 35

r dr dΦ

Question 36

How are cylindrical polar and Cartesian coordinates related?

Answer 36

x = rcos(Φ)
y = rsin(Φ)
z = z
0 =< r =< infinity
0 =< Φ < 2π
-infinity < z < infinity

Question 37

Volume element for cylindrical polar

Answer 37

r dr dΦ dz

Question 38

How are spherical polar and Cartesian coordinates related?

Answer 38

x = rsin(θ)cos(Φ)
y = rsin(θ)sin(Φ)
z = rcos(θ)
r >= 0
0 =< θ =< π
0 =< Φ =< 2π

Question 39

Volume element for spherical polar

Answer 39

r^2 sin(θ) dr dθ dΦ

Question 40

Distance of point A, vector a, from plane r.n̂ = l

Answer 40

distance = |l - (a.n̂)|