Geometry Flashcards
Define vector
An object with magnitude and direction
Define the standard basis of R^n
The standard basis of R^n is:
(1,0,…,0), (0,1,0,…,0),…,(0,0,…,1)
Define magnitude
The square root of the sum of the squares of each of the individual components of the vector.
Define the dot product
u.v = the sum of the product of corresponding coefficients in the vectors u and v.
Define the angle between vectors
cosθ = (u.v)/(|u||v|)
Define a median of a triangle
A line connecting a vertex to the midpoint of the opposite side
Define an altitude of a triangle
A line from a vertex meeting the opposite side orthogonally
Define a perpendicular bisector of a line
A line which divides an edge with a perpendicular line through the mid point
Define the centroid of a triangle
The point where all 3 medians intersect
Define the orthocentre of a triangle
The point where all three altitudes intersect
Define the circumcentre of a triangle
The point where all 3 perpendicular bisectors intersect
Define the Euler line
A line on which the centroid, orthocentre and circumcentre all lie
Define a vector projection
The vector projection of u onto v is the orthogonal projection of u onto a line parallel to v.
Give the parametric form of a line
r(λ) = p + λa, where p is a position vector and a is a direction vector
Give the cartesian form of a line in 3D
(x-p)/a = (y-q)/b = (z-r)/c, deduced from setting all components of the parametric form equal to λ.
Give the parametric form of a plane
r(λ,μ) = p + λa + μb, where a and b are direction vectors which are not multiples of one another and p is a position vector
Give a simplified form of a plane in 3D
r.n=c, where n is a vector orthogonal to the plane and c is a constant.
Define the vector product
Let u = (u1, u2, u3) and v = (v1, v2, v3), the vector product is the determinant of the matrix given by entries: i, j, k, u1, u2, u3, v1, v2, v3.
Define the scalar triple product
[u, v, w] = u.(v^w)
Define the vector triple product
Given u, v, w ∈ R^3, the vector triple product is u^(v^w)
Define the scalar quadruple product
Given a,b,c,d ∈ R^3, the scalar quadruple product is (a^b).(c^d)
Give the cross product equation of a line
r^a = b, which is equivalent to r(α) = αa + (a^b)/(a\^2), so a is the direction vector and (a^b)/(a\^2) is a point on the plane
Define eccentricity
If P is the set of points on a conic, D is a directrix and F is a focus, e = |PF|/|PD|
Give the normal form of an ellipse
x^2/a^2 + y^2/b^2 = 1
Give the normal form of a hyperbola
x^2/a^2 - y^2/b^2 = 1
Give the normal form of a parabola
y^2 = 4ax
For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the eccentricity
e = sqrt(1-b^2/a^2)
For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the eccentricity
e = sqrt(1+b^2/a^2)
Give the eccentricity of a circle`
0
Give the eccentricity of a parabola
1
For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the coordinates of the foci
(±ae,0)
For a parabola, y^2 = 4ax, give the coordinates of the focus
(a,0)
For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the coordinates of the focus
(±ae,0)
For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the directrix
x = ±a/e
For a parabola, y^2 = 4ax, give the directrix
x = -a
For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the directrix
x = ±a/e
Give the area of the ellipse, x^2/a^2 - y^2/b^2 = 1
Area = πab
For a parabola, y^2 = 4ax, give the vertex
(0,0)
For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the assymptotes
y = ±bx/a
Define an isometry
A distance preserving map T, where |T(x) - T(y)| = |x - y|
Give the matrix describing a rotation by t.
cost -sint
sint cost
Give the matrix describing a reflection in the line y = tant
cos2t sin2t
sin2t -cos2t
Define an orthonormal basis
A basis where for all v_i, v_j in the basis, v_i and v_j are orthogonal
How can you determine whether a matrix is a rotation?
Determinant of the matrix is 1
What is the angle of rotation of a matrix?
The angle rotated by matrix A is θ such that tr(A) = 1 + 2cosθ
How can you determine whether a matrix is a reflection?
Determinant of the matrix is -1 and the trace is 1
Define angular velocity
If A describes a rotation and (dA/dt)(A^T) =
0 -γ β
γ 0 -α
-β α 0
Then the angular elocity is (α, β, γ)^T
Define a smooth surface
A smooth surface has x, y, z with continuous derivatives of all orders
Define a tangent plane
The tangent plane at p is the plane containing p, spanned by the vectors ∂p/∂u and ∂p/∂v
Define the normal of a surface
If r is a surface parameterised by u and v, the normal at p is a vector through p with direction ∂r/∂u and ∂r/∂v at p.
Define the arc length of a surface
If γ is parameterised by t, then the length is the integral of |γ’(t)| with respect to t.
Define a geodesic
A curve γ(s) such that γ’‘(s)^n(s) = 0 for all points s.
