Geometry

Question 1

Define vector

Answer 1

An object with magnitude and direction

Question 2

Define the standard basis of R^n

Answer 2

The standard basis of R^n is:
(1,0,…,0), (0,1,0,…,0),…,(0,0,…,1)

Question 3

Define magnitude

Answer 3

The square root of the sum of the squares of each of the individual components of the vector.

Question 4

Define the dot product

Answer 4

u.v = the sum of the product of corresponding coefficients in the vectors u and v.

Question 5

Define the angle between vectors

Answer 5

cosθ = (u.v)/(|u||v|)

Question 6

Define a median of a triangle

Answer 6

A line connecting a vertex to the midpoint of the opposite side

Question 7

Define an altitude of a triangle

Answer 7

A line from a vertex meeting the opposite side orthogonally

Question 8

Define a perpendicular bisector of a line

Answer 8

A line which divides an edge with a perpendicular line through the mid point

Question 9

Define the centroid of a triangle

Answer 9

The point where all 3 medians intersect

Question 10

Define the orthocentre of a triangle

Answer 10

The point where all three altitudes intersect

Question 11

Define the circumcentre of a triangle

Answer 11

The point where all 3 perpendicular bisectors intersect

Question 12

Define the Euler line

Answer 12

A line on which the centroid, orthocentre and circumcentre all lie

Question 13

Define a vector projection

Answer 13

The vector projection of u onto v is the orthogonal projection of u onto a line parallel to v.

Question 14

Give the parametric form of a line

Answer 14

r(λ) = p + λa, where p is a position vector and a is a direction vector

Question 15

Give the cartesian form of a line in 3D

Answer 15

(x-p)/a = (y-q)/b = (z-r)/c, deduced from setting all components of the parametric form equal to λ.

Question 16

Give the parametric form of a plane

Answer 16

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

Question 17

Give a simplified form of a plane in 3D

Answer 17

r.n=c, where n is a vector orthogonal to the plane and c is a constant.

Question 18

Define the vector product

Answer 18

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.

Question 19

Define the scalar triple product

Answer 19

[u, v, w] = u.(v^w)

Question 20

Define the vector triple product

Answer 20

Given u, v, w ∈ R^3, the vector triple product is u^(v^w)

Question 21

Define the scalar quadruple product

Answer 21

Given a,b,c,d ∈ R^3, the scalar quadruple product is (a^b).(c^d)

Question 22

Give the cross product equation of a line

Answer 22

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

Question 23

Define eccentricity

Answer 23

If P is the set of points on a conic, D is a directrix and F is a focus, e = |PF|/|PD|

Question 24

Give the normal form of an ellipse

Answer 24

x^2/a^2 + y^2/b^2 = 1

Question 25

Give the normal form of a hyperbola

Answer 25

x^2/a^2 - y^2/b^2 = 1

Question 26

Give the normal form of a parabola

Answer 26

y^2 = 4ax

Question 27

For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the eccentricity

Answer 27

e = sqrt(1-b^2/a^2)

Question 28

For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the eccentricity

Answer 28

e = sqrt(1+b^2/a^2)

Question 29

Give the eccentricity of a circle`

Answer 29

0

Question 30

Give the eccentricity of a parabola

Answer 30

1

Question 31

For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the coordinates of the foci

Answer 31

(±ae,0)

Question 32

For a parabola, y^2 = 4ax, give the coordinates of the focus

Answer 32

(a,0)

Question 33

For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the coordinates of the focus

Answer 33

(±ae,0)

Question 34

For an ellipse, x^2/a^2 + y^2/b^2 = 1, give the directrix

Answer 34

x = ±a/e

Question 35

For a parabola, y^2 = 4ax, give the directrix

Answer 35

x = -a

Question 36

For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the directrix

Answer 36

x = ±a/e

Question 37

Give the area of the ellipse, x^2/a^2 - y^2/b^2 = 1

Answer 37

Area = πab

Question 38

For a parabola, y^2 = 4ax, give the vertex

Answer 38

(0,0)

Question 39

For a hyperbola, x^2/a^2 - y^2/b^2 = 1, give the assymptotes

Answer 39

y = ±bx/a

Question 40

Define an isometry

Answer 40

A distance preserving map T, where |T(x) - T(y)| = |x - y|

Question 41

Give the matrix describing a rotation by t.

Answer 41

cost -sint
sint cost

Question 42

Give the matrix describing a reflection in the line y = tant

Answer 42

cos2t sin2t
sin2t -cos2t

Question 43

Define an orthonormal basis

Answer 43

A basis where for all v_i, v_j in the basis, v_i and v_j are orthogonal

Question 44

How can you determine whether a matrix is a rotation?

Answer 44

Determinant of the matrix is 1

Question 45

What is the angle of rotation of a matrix?

Answer 45

The angle rotated by matrix A is θ such that tr(A) = 1 + 2cosθ

Question 46

How can you determine whether a matrix is a reflection?

Answer 46

Determinant of the matrix is -1 and the trace is 1

Question 47

Define angular velocity

Answer 47

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

Question 48

Define a smooth surface

Answer 48

A smooth surface has x, y, z with continuous derivatives of all orders

Question 49

Define a tangent plane

Answer 49

The tangent plane at p is the plane containing p, spanned by the vectors ∂p/∂u and ∂p/∂v

Question 50

Define the normal of a surface

Answer 50

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.

Question 51

Define the arc length of a surface

Answer 51

If γ is parameterised by t, then the length is the integral of |γ’(t)| with respect to t.

Question 52

Define a geodesic

Answer 52

A curve γ(s) such that γ’‘(s)^n(s) = 0 for all points s.