Geometry Flashcards
What is a vector?
A list of real numbers
What does Rⁿ mean?
For a given n, we denote the set of all vectors with n co-ordinates as Rⁿ, and
often refer to Rⁿ as n-dimensional co-ordinate space or simply as n-dimensional space
What is the standard/canconical basis of Rⁿ
e₁, e₂, ..., eₙ (1, 0, ..., 0) (0, 1, ...., 0) .... (0, 0, ...., 1)
What is the triangle inequality?
u + v | ≤ |u| + |v|
Prove the triangle inequality
Pg7
What is the dot/scalar/Euclidean inner product?
u . v = u₁v₁ + u₂v₂ + … + uₙvₙ
Let u, v, w be vectors in Rn and let λ be a real number. Then
(a) commutivity
(b) (λu) · v = [ ].
(c) (u + v) · w = [ ].
(d) [ ] . [ ] = |u|² ≥ 0 and u · u = 0 if and only if [ ].
(e) Cauchy-Schwarz Inequality
|u · v| ≤ |u| |v| (1.3)
with equality when one of u and v is a multiple of the other.
Let u, v, w be vectors in Rn and let λ be a real number. Then (a) u · v = v · u. (b) (λu) · v = λ(u · v). (c) (u + v) · w = u · w + v · w. (d) u · u = |u| 2 0 and u · u = 0 if and only if u = 0. (e) Cauchy-Schwarz Inequality |u · v| |u| |v| (1.3) with equality when one of u and v is a multiple of the other.
|u| = ??
In terms of dot product
|u| = √ u.u
What is the angle between two vectors u and v?
cos⁻¹(u.v/(|u| |v|))
two vectors u and v are perpendicular if and only if [ ] = 0
u . v = 0
What is the cosine rule?
a² = b² + c² - 2bccosα
Prove the cosine rule
proof pg 9
What is Thales theorem?
Let A and B be points at opposite ends of the diameter of a circle, and let P be a third point. Then ∡AP B is a right angle if and only if P also lies on the circle.
Prove Thales theorem
Pg 9 (end)
The medians of a triangle are concurrent at its [ ]
Centroid
Prove:
The medians of a triangle are concurrent at its centroid
pg10
Describe the parametric form of a line
Let p, a be vectors in Rⁿ with a ≠ 0. Then the equation r(λ) = p + λa, where λ is a real number, is the equation of the line through p, parallel to a. It is said to be in parametric form, the parameter here being λ. The parameter acts as a co-ordinate on the line, uniquely associating to each point on the line a value of λ.
When are two vectors linearly independent?
We say that two vectors in Rⁿ are linearly independent, or just simply independent, if neither is a scalar multiple of the other.
In particular, this means that both vectors are non-zero.
Two vectors which aren’t independent are said to be [ ]
linearly dependent
What is the parametric form of a plane?
Let p, a, b be vectors in Rⁿ with a, b independent.
Then
r(λ, µ) = p + λa + µb where λ, µ are real numbers is the equation of the plane through p parallel to the vectors a, b. The parameters λ, µ act as co-ordinates in the plane, associating to each point of the plane a unique ordered pair (λ, µ) for if
p + λ₁a + µ₁b = p + λ₂a + µ₂b
then (λ₁ − λ₂) a = (µ₂ − µ₁
) b so that λ₁ = λ₂ and µ₁ = µ₂
by independence
What is the Cartesian Equation of a Plane in R³?
A region Π of R³ is a plane if
and only if it can be written as r · n = c
where r = (x, y, z), n = (n₁, n₂, n₃) ≠ 0 and c is a real number. In terms of the co-ordinates
x, y, z this equation reads
n₁x + n₂y + n₃z = c
The vector n is normal (i.e. perpendicular) to the plane
Prove the Cartesian Equation of the plane in R³
Proof pg 14
What is the vector/cross product?
u ∧ v =
| i j k |
| u₁ u₂ u₃|
| v₁ v₂ v₃ |
For u, v in R³, we have
|u ∧ v|² =
|u ∧ v|² = |u|²|v|² - (u.v)²
When does u ∧ v = 0?
In particular u ∧ v = 0 if and only if u and v are linearly dependent.
|u ∧ v| = |u| |v| [ ]
For u, v in R³ we have |u ∧ v| = |u| |v|sin θ where θ is the smaller angle between u and v
Prove that |u ∧ v| = |u| |v|sin θ
|u ∧ v|² = |u|²|v|² - (u.v)² = |u|²|v|²(1 - cos²θ) = |u|²|v|² sin²θ
For u, v in R³ then |u ∧ v| equals the area of the parallelogram with vertices….
0, u, v, u+v
For u, v, w in R³, and reals α, β we have:
(αu + βv) ∧ w = [ ]
(αu + βv) ∧ w = α(u ∧ w) + β(v ∧ w)
For u, v, w in R³, and reals α, β we have:
u ∧ v = − [ ]
u ∧ v = −v ∧ u.
For u, v, w in R³, and reals α, β we have:
u ∧ v is perpendicular to [ ]
u ∧ v is perpendicular to both u and v
For u, v, w in R³, and reals α, β we have:
If u, v are perpendicular unit vectors then [ ] is a unit vector
If u, v are perpendicular unit vectors then u ∧ v is a unit vector.
For u, v, w in R³, and reals α, β we have:
If u, v are perpendicular unit vectors then [ ] is a unit vector
If u, v are perpendicular unit vectors then u ∧ v is a unit vector.
Given independent vectors u and v in R³, the plane containing the origin 0 and
parallel to u and v has equation ….
n r · (u ∧ v) = 0
Given independent vectors u and v in R³, the plane containing the origin 0 and
parallel to u and v has equation ….
r · (u ∧ v) = 0
Define the scalar triple product
[u, v, w]
[u, v, w] = u . ( v ∧ w)
[u, v, w] = [ ] [v, w, u] = [ ] [w, u, v] = [ ] [u, w, v] = [ ] [v, u, w] = [ ] [w, v, u]
[u, v, w] = [v, w, u] = [w, u, v] = − [u, w, v] = − [v, u, w] = − [w, v, u]
Note that [u, v, w] = 0 if and only if ….
Note that [u, v, w] = 0 if and only if u, v, w are linearly dependent; this is equivalent to the 3 × 3 matrix with rows u, v, w being singular.
What is the volume of a parallelepiped with vertices 0, u, v, w?
|[u, v, w]|
What is the vector triple product?
Given three vectors u, v, w in R³ we define their vector triple product as
u ∧ (v ∧ w).
u ∧ (v ∧ w) = ??? (In terms of dot and vector product)
u ∧ (v ∧ w) = (u · w)v − (u · v)w
Prove that u ∧ (v ∧ w) = (u · w)v − (u · v)w
proof end pg20
What is the scalar quadruple product?
Given four vectors a, b, c, d in R³, their scalar quadruple product is
(a ∧ b) · (c ∧ d)
(a ∧ b) · (c ∧ d) = ??? (In terms of dot product)
(a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c)
Prove that
a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c
Set e = c ∧ d. Then (a ∧ b) · (c ∧ d) = e · (a ∧ b) = [e, a, b] = [a, b, e] = a · (b ∧ e) = a · (b ∧ (c ∧ d)) = a · ((b · d)c − (b · c)d) [by the vector triple product] = (a · c)(b · d) − (a · d)(b · c).
Let a and b be linearly independent vectors in R³. Then a, b and a∧b form
a basis for R³. This means that for every v in R³ there are unique real numbers α, β, γ such
that v = [ ]
We will refer to α, β, γ as the co-ordinates of v with respect to this basis
v = αa + βb + γa ∧ b
What is the vector equation of a line using vector product?
Let a, b vectors in R³ with a · b = 0 and a ≠ 0. The vectors r in R³ which satisfy r ∧ a = b form the line parallel to a which passes
through the point (a ∧ b)/ |a|²
det AB =
det AB = det A det B
a square matrix is singular if and only if [ ].
a square matrix is singular if and only if has zero determinant.
What is the equation for a double cone?
x² + y² = z²cot²α
What are the four types of conic sections?
Circle
Ellipse
Parabola
Hyperbola
Let D be a line, F a point not on the line D and e > 0. Then the conic with directrix D and focus F and eccentricity e, is the set of points P (in the plane containing
F and D) which satisfy the equation ….
|P F| = e|PD|
where |P F| is the distance of P from the focus and |PD| is the distance of P from the directrix.
That is, as the point P moves around the conic, the shortest distance from P to the line D is in constant proportion to the distance of P from the point F.
Conics
If 0 < e < 1 then the conic is called an [ ]
ellipse
Conics
If e = 1 then the conic is called a [ ]
parabola
Conics
If e > 1 then the conic is called a [ ]
hyperbola
What is the equation of an ellipse? Define each variable
xy-co-ordinates
x²/a² + y²/b² = 1 0 < b < a
where
a = ke/(1-e²) and b = ke/√ (1-e²)
and e = √ (1 - b²/a²)
In the xy-co-ordinates,
where is the focus and directrix of an ellipse?
The focus F is at (ae, 0) and the directrix D is the line x = a/e
What is the area of an ellipse in xy-co-ordinates?
πab
a = ke/(1-e²) and b = ke/√ (1-e²)
and e = √ (1 - b²/a²)
How can an ellipse be parametrized using trig?
x = a cost, y = b sin t, 0 ≤ t < 2π
How else can an ellipse be parametrized (no trig)?
x = a(1 - t²)/(1 + t²)
y = b(2t)/(1 + t²)
(−∞ < t < ∞)
What is the equation of a hyperbola? Define each variable
xy-co-ordinates
x²/a² - y²/b² = 1 0 < a, b
where
a = ke/(e² - 1) and b = ke/√ (e² - 1)
and e = √ (1 + b²/a²)
In the xy-co-ordinates,
where is the focus and directrix of a hyperbola?
The focus F is at (ae, 0) and the directrix D is the line x = a/e
What are the equations for the asymptotes of a hyperbola?
The lines ay = ±bx are known as the asymptotes of the hyperbola; these are, in a sense, the tangents to the hyperbola at its two ‘points at infinity’.
What is a right hyperbola?
When e = √2 (i.e. when a = b) then the asymptotes are perpendicular and C is known as a right hyperbola.
How can a hyperbola be parametrized using hyperbolic trig?
x = ±a cosh t y = b sinh t
−∞ < t < ∞)
How else can a hyperbola be parametrized (no trig)?
x = a(1 + t²)/(1 - t²)
y = b(2t)/(1 - t²)
(t ≠ ±1)
What is the normal form of a parabola?
y² = 4ax
In the xy-co-ordinates,
where is the focus and directrix of a parabola?
The focus is the point (a, 0) and the directrix is the
line x = −a
Where is the vertex of a parabola?
The vertex of the parabola is at (0, 0)
What is the parametrized form of a parabola?
(x, y) = (at², 2at) where −∞ < t < ∞.
What is the eccentricity of a circle?
e = 0
Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an [ ] with foci at [ ] and [ ]
Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an ellipse with foci at A and B
Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an ellipse with foci at A and B
prove it
If we consider the ellipse with foci A and B and a point P on the ellipse we have
|AP| + |P B| = |F₁P| + |P F₂|
= e |D₁P| + e |D₂P|
= e |D₁D₂|
where D₁D₂ is the perpendicular distance between the two directrices. Thus the value is constant on any ellipse with foci A and B and will take different values for different ellipses as the value of |AP| + |P B| increases as P moves right along the line AB.
Show that the curve
x² + xy + y² = 1
is an ellipse
A rotation about the origin in R² by θ anti-clockwise takes the form
X = x cos θ + y sin θ, Y = −x sin θ + y cos θ;
x = X cos θ − Y sin θ, y = X sin θ + Y cos θ.
Writing c = cos θ and s = sin θ, for ease of notation, our equation becomes
(Xc − Y s)² + (Xc − Y s)(Xs + Y c) + (Xs + Y c)² = 1
which simplifies to
(1 + cs)X² + (c² − s²)XY + (1 − cs)Y²= 1.
So if we wish to eliminate the xy-term then we want
cos 2θ = c² − s²= 0
which will be the case when θ = π/4, say. For this value of θ we have c = s = 1/√2 and our equation has become
3/2 X² + 1/2 Y² = 1.
The solutions of the equation
Ax² + Bxy + Cy² = 1
where A, B, C are real constants, such that A, B, C are not all zero, form one of the following
types of loci: ….
pes of loci:
Case (a): If B² − 4AC < 0 then the solutions form an ellipse or the empty set.
Case (b): If B² − 4AC = 0 then the solutions form two parallel lines or the empty set.
Case (c): If B² − 4AC > 0 then the solutions form a hyperbola.
Ax² + Bxy + Cy² + Dx + Ey + F = 0
Describe the different cases for the loci of solutions
Case (a): If B² − 4AC < 0 then the solutions form an ellipse, a single point or the empty set.
Case (b): If B² − 4AC = 0 then the solutions form a parabola, two parallel lines, ansingle line or the empty set.
Case (c): If B² − 4AC > 0 then the solutions form a hyperbola or two intersecting
lines.
What is an isometry?
An isometry T from Rⁿ to Rⁿ is a distance-preserving map. That is:
|T(x) − T(y)| = |x − y| for all x, y in Rⁿ
Rotations, [ ] and [ ] are all examples of isometries
reflections
translations
Describe the map Rθ, which denotes rotation by θ anti-clockwise about the origin
Rθ = ( cosθ - sinθ )
( sinθ cosθ )
Describe the map Sθ, which is reflection in the line
y = x tan θ
Sθ = ( cos2θ sin2θ )
( sin2θ -cos2θ )
A square real matrix A is said to be orthogonal if ….
A⁻¹ = Aᵀ
The n × n orthogonal matrices are the linear [ ] of Rⁿ
isometries
Let A be an n × n orthogonal matrix and b ∈ Rⁿ. The map
T(x) = Ax + b
is an [ ] of Rⁿ
isometry
Let A and B be n × n orthogonal matrices
Is AB orthogonal?
Yes
Let A and B be n × n orthogonal matrices
Is A⁻¹ orthogonal?
Yes
Let A and B be n × n orthogonal matrices
What is detA = ???
detA = ±1
We say that n vectors v₁, v₂, …, vₙ are a basis for Rⁿ if …
if for every v ∈ Rⁿ there exist unique real numbers α₁, α₂, …, αₙ such that
v = α₁v₁ + α₂v₂ + · · · + αₙvₙ.
What is an orthonormal basis?
A basis v₁, v₂, …, vₙ for Rⁿ is orthonormal if
vᵢ · vⱼ = δᵢⱼ = { 1 if i = j
{ 0 if i ≠ j
[ ] orthonormal vectors in Rⁿ
form a basis. (In fact, orthogonality alone is sufficient to guarantee linear [ ] .)
n
independence
If v₁, v₂, …, vₙ are an orthonormal basis for Rⁿ and x = α₁v₁ + α₂v₂ + · · · + αₙvₙ
then note that
αᵢ = [ ]
αᵢ = x · vᵢ
An n × n matrix A is orthogonal if and only if …
An n × n matrix A is orthogonal if and only if its columns form an orthonormal basis for Rⁿ. The same result hold true for the rows of A.
Let A be an orthogonal 3 × 3 matrix, and x, y be column vectors in R³.
(a) If det A = 1 then A(x ∧ y) = [ ]
(b) If det A = −1 then A(x ∧ y) = [ ]
Let A be an orthogonal 3 × 3 matrix, and x, y be column vectors in R³
(a) If det A = 1 then A(x ∧ y) = Ax ∧ Ay.
(b) If det A = −1 then A(x ∧ y) = −Ax ∧ Ay.
What is the Spectral Theorem (Finite dimensional case)?
Let A be a square real symmetric matrix (so Aᵀ = A). Then there is an orthogonal matrix P such that PᵀAP is diagonal.
(Classifying 3 × 3 orthogonal matrices) Let A be a 3 ×3 orthogonal matrix.
If det A = 1 then A is a ….
If det A = 1 then A is a rotation of R³ about some axis by an angle θ where
trace A = 1 + 2 cos θ.
(Classifying 3 × 3 orthogonal matrices) Let A be a 3 ×3 orthogonal matrix.
If det A = −1 and trace A = 1 ….
If det A = −1 and trace A = 1 then A is a reflection of R³. The converse also holds.
Let S be an isometry from Rn
to Rⁿ such that S(0) = 0. Then
(a) |S(v)| = [ ] for any v in Rⁿ and [ ] = u · v for any u, v in Rⁿ.
(b) If v1, …, vn is an orthonormal basis for Rⁿ then so is [ ]
(c) There exists an orthogonal matrix A such that S(v) = [ ] for each v in Rⁿ
Let S be an isometry from Rn
to Rⁿ such that S(0) = 0. Then
(a) |S(v)| = |v| for any v in Rⁿ and S(u) · S(v) = u · v for any u, v in Rⁿ.
(b) If v1, …, vn is an orthonormal basis for Rⁿ then so is S(v1), …, S(vn).
(c) There exists an orthogonal matrix A such that S(v) = Av for each v in Rⁿ
Let T be an isometry from Rn
to Rⁿ. Then there is a orthogonal matrix A and a column vector b such that T(v) = [ ] for all v.
Further A and b are [ ] in this regard.
Av + b
unique
Euler Angles
Let R denote an orthogonal 3 × 3 matrix with det R = 1 and
let
R (i, θ) =
R (i, θ) = ( 1 0 0 )
( 0 cosθ - sinθ )
( 0 sinθ cosθ )
Let R denote an orthogonal 3 × 3 matrix with det R = 1 and
let
R (j, θ) =
R (j, θ) = ( cosθ 0 -sinθ )
( 0 1 0 )
( sinθ 0 cosθ )
Let’s suppose that the fixed-in-body and fixed-in-space axes share a common origin throughout the motion. Then at a time t the moving axes will be a rotation A(t) from the fixed axes.
We then have the equation
A(t)A(t)ᵀ = I
What does
dA/dt Aᵀ = ???
dA/dt Aᵀ = (0 −γ β )
(γ 0 −α )
(−β α 0 )
for some α, β, γ ∈ R (which of course may still depend on t)
Let’s suppose that the fixed-in-body and fixed-in-space axes share a common origin throughout the motion. Then at a time t the moving axes will be a rotation A(t) from the fixed axes.
We then have the equation
A(t)A(t)ᵀ = I
Define the angular velocity of a body at time t
ω(t) = (α, β, γ)ᵀ
What does
( x )
ω ∧ ( y ) = ???
( z )
( x )
ω ∧ ( y ) = dA/dt Aᵀ (x, y, z)ᵀ
( z )
Rotating frames
Let r(t) be the position vector at time t of a point fixed in the body — this is the point’s
position vector relative to fixed-in-space axes.
Then r(t) = A(t) [ ]
Differentiating this we
find ….
Then r(t) = A(t)r(0) dr/dt = dA/dr r(0) = (dA/dt Aᵀ) (Ar(0)) = (0 −γ β ) (γ 0 −α ) r(t) (−β α 0 )
= ω(t) ∧ r(t)
Give the Cartesian equation for the following parametrized surfaces in R³:
Sphere
x² + y² + z² = a²
Give the Cartesian equation for the following parametrized surfaces in R³:
Ellipsoid
x²/a² + y²/b² + z²/c²
Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperboloid of Two Sheets
x²/a² - y²/b² - z²/c² = 1
Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperboloid of One Sheet
x²/a² + y²/b² - z²/c² = 1
Give the Cartesian equation for the following parametrized surfaces in R³:
Paraboloid
z = x² + y²
Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperbolic Paraboloid
z = x² - y²
Give the Cartesian equation for the following parametrized surfaces in R³:
Cone
z² = x² + y² with z ≥ 0
Define arc length given a parametrized curve r: [a, b] → R³
arc length = ₐ∫ᵇ |r’(t)| dt
What is a smooth parametrized surface?
A smooth parametrized surface is a map r, known as the parametrization
r : U → R³
given by r (u, v) = (x (u, v), y (u, v), z (u, v)) from an (open) subset U ⊆ R² to R³ such that:
• x, y, z have continuous partial derivatives with respect to u and v of all orders;
• r is a bijection from U to r(U) with both r and r⁻¹ being continuous;
• (smoothness condition) at each point the vectors
rᵤ = ∂r/∂u and rᵥ = ∂r/∂v
are linearly independent (i.e. are not scalar multiples of one another).
Define Cylindrical Polar Co-ordinates)
These are given by
x = r cos θ, y = r sin θ, z = z,
with r > 0, −π < θ < π, z ∈ R.
Define Spherical Polar Co-ordinates
These are given by
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,
with r > 0, 0 < θ < π, −π < φ < π.
What is the parametrization of a cone using cylindrical
polar co-ordinates?
r(θ, z) = (z cos θ, z sin θ, z) 0 < θ < 2π, z > 0
Note that the parametrization misses one meridian of the cone
Let r: U → R³ be a smooth parametrized surface and let p be a point on the
surface
What is the tangent plane to r(U) at p?
The plane containing p and which is parallel to the vectors
rᵤ(p) = ∂r/∂u (p)
rᵥ(p) = ∂r/∂v (p)
Let r: U → R³ be a smooth parametrized surface and let p be a point on the
surface
Is the tangent plane to r(U) at p well defined? And why?
Because rᵤ and rᵥ are independent the tangent plane
is well-defined.
Any vector in the direction
∂r/∂u (p) ∧ ∂r/∂v (p)
is said to be [ ] to the surface at p. Thus there are two [ ] of length one.
normal
unit normals
What is the parametrization of a sphere using spherical polar co-ordinates?
Spherical polar co-ordinates give a natural parametrization for
the sphere x² + y² + z² = a² with
r (φ, θ) = (a sin θ cos φ, a sin θ sin φ, a cos θ), −π < φ < π, 0 < θ < π.
We can form a surface of revolution by rotating the graph y = f(x), where f(x) > 0, about the x-axis. There is then a fairly natural parametrization for the surface of revolution with cylindrical polar co-ordinates:
r(x , θ) = ???
r(x, θ) = (x, f(x) cos θ, f(x) sin θ) − π < θ < π, a < x < b
What is a catenoid and what is its parametrization?
The catenoid is the surface of revolution formed by rotating the curve y =
cosh x, known as a catenary, about the x-axis. So we can parametrize it as
r(x, θ) = (x, cosh x cos θ, cosh x sin θ), −π < θ < π, x ∈ R
What is a helicoid and what is its parametrization?
The helicoid is formed in a “propeller-like” fashion by pushing the x-axis up the z-axis while
spinning the x-axis at a constant angular velocity. So we can parametrize it as
s(X, Z) = (X cosZ, X sinZ, Z)
Let r : U → R³ be a smooth parametrized surface.
What is the (surface) area of r(U)?
∫∫ᵤ |∂r/∂u ∧ ∂r/∂v| du dv
We will often write
dS = [ ] du dv
to denote an infinitesimal part of surface area
∂r/∂u ∧ ∂r/∂v|
The surface area of r (U) is independent of the choice of [ ]
parametrization
Let z = f (x, y) denote the graph of a function
f defined on a subset S of the xy-plane
What is the surface area of the graph?
∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy
Let z = f (x, y) denote the graph of a function
f defined on a subset S of the xy-plane
Show that the graph has surface area ∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy
We can parametrize the surface as r (x, y) = (x, y, f (x, y)) (x, y) ∈ S. Then rₓ ∧ rᵧ = (-fₓ, -fᵧ, 1) Hence the graph has surface area ∫∫ₛ |rₓ ∧ rᵧ| dx dy = ∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy
A surface S is formed by rotating the graph of
y = f(x) a < x < b,
about the x-axis. (Here f(x) > 0 for all x.) The surface area of S equals ?????
Area(S) = 2π ₓ₌ₐ∫ˣ⁼ᵇ f(x) ds/dx dx
Isometries preserve [ ]
Area
Angles (between curves)
What is the volume of a tetrahedron with vertices 0, u, v, w?
1/6 |[u, v, w]|
