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.
