Vectors

What is a vector?

A directed line segment characterised by two ordered points, P and Q, PQ^→

P is the initial point and Q is the terminal point

The length of PQ^→ is…

… the distance between P and Q, denoted ||PQ^→||.

It is also called the norm of PQ^→

If P and Q coincide so P=Q then the vector PQ^→ is called…

… the zero vector/segment, denoted 0.

PQ^→ = 0 iff…

… ||PQ^→|| = 0

Two directed line segments PQ^→ and RS^→ are said to be parallel (or collinear) if…

… they lie on parallel straight lines

The zero segment is parallel to…

… any vector

PQ^→ and RS^→ are equivalent/equal if…

… they are collinear, have the same magnitude and point in the same direction

If a = PQ^→ and b = QR^→, then PR^→ = c = ?

a + b

For any real number, λ, the scalar multiplication of a by λ produces the vector b = λa such that…

(three things)

• b is collinear to a

• ||b|| = |λ|||a||

• b and a have different directions if 0>λ and the same if λ>0

Given a non-zero vector a≠0, the unit vector is…

… hat(a) = a/||a|| in the direction of a.

What is a position vector?

A position vector is a way of identifying a point as a vector from the origin point, O.

Define linear independence

Let a and b be non-zero, non-parallel vectors and let λ, µ be real numbers.

Then λa + µb = 0 => λ=0 or µ=0.

If a = (a₁, a₂, a₃), then a₁, a₂, a₃ are known as…

… the components of a and are unique.

Write a in terms of the unit vectors

a = a₁i + a₂j + a₃k

If a = (a₁, a₂, a₃), then ||a|| = ?

√(a₁² + a₂² + a₃²)

If a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) then a + b = ?

(a₁+b₁, a₂+b₂, a₃+b₃)

If a = (a₁, a₂, a₃), then λa = ?

(λa₁ + λa₂ + λa₃)

Given two vectors a and b, their dot or scalar product is given by…

… a_^ob = ||a||||b||cosθ where θ is the angle between a and b, 0≤θ≤π

a and b are orthogonal/perpendicular if…

… a_^ob = 0

Component form of the dot/scalar product

a_^ob = a₁b₁ + a₂b₂ + a₃b₃

The dot/scalar product is…

… associative and commutative

Define ||a|| in terms of the dot product

||a|| = √(a_^oa)

If x_^oc = 0 ∀x, then…

… c = 0

If x_^oa = x_^ob ∀x, then…

a = b

If x_^oa = x_^ob for some x, then…

x is orthogonal to a - b

What is orthogonal projection?

We can construct b_// and b_⊥ such that:

• b_// + b_⊥ = b

• b_// is parallel to a

• b_⊥ is perpendicular to a

b_// = ?

(b_^oa)a/||a||²

b_⊥ = ?

b - (b_^oa)a/||a||²

Given two vectors a and b, which are at an angle θ to each other, their cross product c, denoted c = axb is a vector which fulfills:

(three things)

• c is orthogonal to a and b

• ||c|| = ||a||||b||sinθ (0≤θ≤π)

• If a//b then axb = 0 (otherwise form a right angle system)

The cross product is…

… anticommutative and associative

a x a = ?

0

i x j = ?

k

j x k = ?

i

k x i = ?

j

Component form of the cross product

a x b = (a₂b₃-a₃b₂)i + (a₃b₁-a₁b₃)j + (a₁b₂-a₂b₁)k

(i.e. the determinant of the matrix [ijk,a,b])

Given three vectors a, b, c, their scalar triple product is the scalar…

… [a, b, c] = a _^o (b x c)

Component form of the scalar triple product

[a, b, c] = a₁(b₂c₃-b₃c₂) + a₂(b₃c₁-b₁c₃) + a₃(b₁c₂-b₂c₁)

(i.e. the determinant of the matrix [a,b,c])

This also calculates the volume of a parallelopiped

Three vectors are coplanar if…

… there exists scalars λ, µ, v, s.t. λa + µb + vc = 0

Then a, b, c are linearly dependent and lie in the same plane

a, b, c are coplanar is equivalent to…

… [a, b, c] = 0

[a, b, c] is equivalent to its…

… cyclic permutations

[a, b, c+d] = ?

[a, b, c] + [a, b, d]

λ[a, b, c] = ?

[λa, b, c]

For a, b, c the vector triple product is…

… a x (b x c) = (c_^oa)b - (b_^oa)c

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

0

A point, P, with position vector r lies on a line L, with position vector l if…

… l is parallel to r-a where a is the position vector of another point on the line.

What is the parametric equation of a straight line?

r = a + λl

What is another equation for a straight line?

(r-a) x l = 0

Two non-parallel lines r = a + λl and r = b +µm intersect iff…

… [a, l, m] = [b, l, m]

How do you find the shortest distance between two straight lines?

Find the formula for c and d on lines L₁ and L₂ such that ||c-d|| is the shortest distance between the two lines.

If c = b + µm and d = a + λl,
µ = [a-b, l, n] / [m, l, n] and λ = [b-a, m, n] / [l, m, n]

What is the parametric equation of a plane?

r = a + λl + µm where a is a position vector and l and m are non-parallel vectors in the plane

What is the equation of the plane relating to the normal?

r _^o n = d, or in component form: ax + by + cz = d

If n is the unit normal, d such that r_^on = d is…

… the perpendicular distance from the plane to the origin

How do you find the intersection of a line and a plane?

Substitute the equation of the line into the equation of the plane and solve for λ

How do you reflect a point in a plane?

Create a line using the point and the normal to the plane, substitute its equation into the equation for the plane and find λ

The reflected point is found by substituting 2λ into the created line equation

What’s the parametric equation for a circular helix?

r(t) = acos(t)i + asin(t)j + ctk, where a is the radius of the circle

r(t) → r⁰ as t → t₀ iff…

… ||r(t) - r⁰|| → 0 as t → t₀

(this is true iff x(t) → x⁰, y(t) → y⁰, z(t) → z⁰ where r(t) = (x(t), y(t), z(t))

Curve r(t) is continuous if…

… ∀t₀, r(t) → r(t₀) as t → t₀

Given r(t), the derivative r'(t) is…

… dr/dt := lim_h→0 [r(t+h)-r(t)/h], provided this limit exists (and so does dx/dt, dy/dt and dz/dt)

Write r' in vector form

r' = idx/dt + jdy/dt + kdz/dt

Assuming r'(t) ≠0 is a tangent vector to the curve c, it…

… points in the direction of increasing t and is precisely the velocity

d/dt(r₁_^or₂) = ?

r₁'_^or₂ + r₁_^or₂'

d/dt(r₁xr₂) = ?

r₁'xr₂ + r₁xr₂'

dr/dt = ?

1/r(r_^or') where r = ||r||

If r is a vector of constant length r, …

r_^or' = 0, so radius and tangent vector are orthogonal

∫^b_a r(t) dt = ?

i∫^b_a x(t) dt + j∫^b_a y(t) dt + k∫^b_a z(t) dt``

Arc length, s(t) = ?

∫^t_t₀ √[(dx/dτ)²+(dy/dτ)²+(dz/dτ)²] dτ

With r'(t) ≠0, hat(T)(t) = ?

r'(t)/||r'(t)|| = dr/ds

The normal vector is N = ?

d/dt(hat(T))

What is the principal unit normal?

hat(N) = hat(T')/||hat(T')||

Define curvature

K = ||dhat(T)/ds|| = ||d²r/ds²|| = ||dhat(T)/dt|| ÷ ||r'||

What is the radius of curvature?

a = 1/K

What is the binormal?

hat(B) = hat(T) x hat(N)

Suppose we have curve C given by r = (x(t), y(t), z(t)), b>t>a.

The vector line integral I of F(r) is given by…

… I = ∫^b_a F_^or' dt = ∫^t₁_t₀ F_^odr

(We can see I as the Work Done)

Let p(r) be a scalar function. The scalar line integral I of ρ(r) is given by…

… I = ∫_c ρ(r) ds = ∫^ba ρ(r(t))||r'(t)|| dt

The path of a ball rolling down a mountain given by vector…

… [∂f/∂x, ∂f/∂y, 1]

At any point (x,y) where the partial derivatives exists, we define the vector ∇f(x,y) (aka grad/del/nabla f(x,y)) by the expression…

… ∇(f_x, f_y) = (∂f/∂x, ∂f/∂y) = f_x(x,y)i + f_y(x,y)j

If c∈R, we define a level curve of points (x,y) for which f(c,y) = c. The contour lines are formed by the set…

… C = {(x,y): f(x,y) = c}, the points (x,y,c) form contour lines of points at height c

The directional derivative of the function f along the curve C is given by…

… dg/dt = r'_^o(f_x, f_y)

C is a straight line in the direction of u = (u,v) through (a,b).

The directional derivative for this case is D_u = ?

uf_x + vf_y = u^_o∇f

If (x,y) is differentiable at (a,b) and ∇f(a,b)≠0 then ∇f(a,b) is…

… a vector normal to the level curve (contour) of f(x,y) at (a,b)

i.e. T_^o∇f = 0

The vector ∇f(a,b) is the direction of the…

… maximum increase / steepest ascent of f at the point (a,b).

(negative is steepest descent)

The velocity v(t) is…

… the rate of change of position (x(t)), x'(t)

It’s also tangential to C so defined as r'

Speed, v, is…

… the magnitude of v

i.e. v = ||v|| = ds/dt = s'

The acceleration vector, a, is…

… the rate of change of velocity, x''(t)

Jerk is…

… the rate of change of acceleration, x'''(t)

r² = ?

x_^ox

rr' = ?

x_^ox'

v_^ov' = ?

vv'

x'' + ω²x = 0 has solution…

… x(t) = Acos(ωt) + Bsin(ωt), a periodic function with period T = 2π/ω

If the initial velocity and position vectors are parallel…

… it is parallel motion, otherwise it will be circular

What equation links e_θ and e_r?

e_θ = a x e_r, where a is a vector for the axis of rotation

For circular motion, x' = ?

θ'a x x

What are the equations linking polar coordinates to cartesian coordinates?

x = rcosθ, y = rsinθ, r² = x² + y², tanθ = y/x

What are the equations for e_θ and e_r in terms of i and j?

e_r = cosθi + sinθj, e_θ = -sinθi + cosθj

What’s an equation for x in terms of e_r?

x = re_r

d/dθ(e_r) = ?

θ'e_θ

d/dθ(e_θ) = ?

-θ'e_r

What is the polar equation for velocity?

x' = r'e_r + rθ'e_θ

(so ||x'|| = √(r'² + r²θ'²))

What is the polar equation for acceleration?

x'' = (r'' - rθ'²)e_r + (rθ'' + 2r'θ')e_θ

Centripetal acceleration

-rθ'²e_r