IB Maths AI Topic 3 Definitions

This page contains our IB Maths AI definitions for topic 3. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Geometry & Trigonometry'.

All Topic 3 Sub-topics 3.1: Fundamentals Of Geometry 3.2: Introduction To Trigonometry 3.3: Applied Trigonometry 3.4: Circles 3.5: Perpendicular Bisectors 3.6: Voronoi Diagrams (HL)3.7: Unit circle (HL)3.8: Geometric transformations (HL)3.9: Vectors (HL)3.10: Vector lines (HL)3.11: Graph theory (HL)3.12: Adjacency matrices (HL)3.13: Algorithms & problems

Adjacency

A way of describing which vertices in a graph are directly connected by an edge.

adjacent

The side next to the chosen acute angle in a right-angled triangle, not including the hypotenuse.

ambiguous case

The situation when two sides and a non-included angle are given for a triangle, where using the sine rule can lead to no triangle, one triangle, or two different triangles because

\sin\theta=\sin\left(\pi-\theta\right)

arc

Part of the circumference of a circle between two points.

arc length

The distance along an arc; for radius

r

and central angle

heta

l=r\theta

when

heta

is in radians, or

l=\frac{\theta}{360}\times2\pi r

when

heta

is in degrees.

bearing

A direction given as an angle measured clockwise from north, written using three figures (for example, east is

090^\circ

bisector

A line that divides a line segment into two equal parts by passing through its midpoint.

Cartesian form

A way to write a complex number as

z=a+bi

with

a,b\in\mathbb{R}

, separating its real and imaginary components.

Chinese postman problem

Finding the shortest closed route that uses every edge of a weighted graph at least once.

Coincident

Describes two lines that are the same line, meaning their direction vectors are parallel and they share at least one common point.

column vector

A way to represent a point $(x,y)$ as $\begin{pmatrix}x\\y\end{pmatrix}$ so a matrix can act on it.

complete

Describes a simple graph in which every pair of distinct vertices is joined by an edge.

cone

A solid with a circular base and a curved surface meeting at a single vertex; for a right cone, $V=\frac{1}{3}\pi r^2h$ where $r$ is the base radius and $h$ is the perpendicular height.

cos

A trigonometric function whose derivative is '

-\sin x

cosine

A trigonometric function whose derivative is

\frac{d}{dx}(\cos x)=-\sin x

cosine rule

A relationship for any triangle linking two sides and the included angle:

c^2=a^2+b^2-2ab\cos C

degree

The highest power of

x

with a non-zero coefficient in a polynomial.

deleted vertex algorithm

Produces a lower bound for the travelling salesman problem by deleting one vertex, finding a minimum spanning tree on the remaining vertices, then adding the two smallest edges that connect the deleted vertex to the tree.

depression

The angle measured downwards from a horizontal line to the line of sight to an object.

determinant

For a

2\times2

matrix

A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}

, the scalar

ad-bc

, which equals the product of the eigenvalues and is used in

\det\left(A-\lambda I\right)

directed

Having edges with a specified direction, so moving from one vertex to another may be possible without the reverse move being possible.

direction vector

A non-zero vector parallel to a line that gives its direction; in $r=a+\lambda b$ , $b$ determines how $r$ changes as $\lambda$ changes.

displacement

The vector that represents the change in position from one point to another, for example $\rightarrow{AB}=\textbf{b}-\textbf{a}$ .

distance

The length of the straight line segment joining two points; in three-dimensional space it can be found using

$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$ .

edge

A connection between two vertices.

elevation

The angle measured upwards from a horizontal line to the line of sight to an object.

enlargement

A transformation about the origin that scales both coordinates by the same factor $k$ , using $\begin{pmatrix}k&&0\\0&&k\end{pmatrix}$

Eulerian circuit

A walk that uses every edge exactly once and starts and ends at the same vertex.

Eulerian trail

A walk that uses every edge exactly once.

even function

A function with symmetry in the

y

-axis, satisfying

f\left(-x\right)=f\left(x\right)

, for example

\cos\left(-x\right)=\cos x

gradient

The coefficient

a

y=ax+b

, giving the predicted change in

y

when

x

increases by

1

Hamiltonian cycle

A path that visits every vertex exactly once and returns to the starting vertex.

Hamiltonian path

A path that visits every vertex exactly once.

hemisphere

Half of a sphere cut by a plane through its centre; its volume is $V=\frac{2}{3}\pi r^3$ .

horizontal stretch

A transformation that multiplies the $x$ -coordinate by a scale factor $k$ while leaving the $y$ -coordinate unchanged, using $\begin{pmatrix}k&&0\\0&&1\end{pmatrix}$

hypotenuse

The longest side in a right-angled triangle, opposite the

90^\circ

angle.

interpolation

Estimating a value at an unknown point using known values at nearby points; in nearest neighbour interpolation, the estimate is taken from the closest site.

Intersecting

Describes two lines that share a common point, found by solving their parametric equations simultaneously for a consistent set of parameter values.

Kruskal’s algorithm

Finds a minimum spanning tree by adding edges in increasing weight order, skipping any edge that would create a cycle, until all vertices are connected.

length

The number of edges used in a walk.

matrices

Rectangular arrays of entries arranged in rows and columns, used to represent and analyse connections in a graph.

midpoint

The point halfway along the line segment joining two points; for $A\left(x_1,y_1,z_1\right)$ and $B\left(x_2,y_2,z_2\right)$ it is $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$ .

minimum spanning tree

A spanning tree with the smallest possible total weight.

nearest neighbour algorithm

Produces a travelling salesman tour by repeatedly going to the nearest unvisited vertex and then returning to the start, giving an upper bound for the optimum tour.

negative reciprocal

The value obtained by changing the sign and inverting a non-zero number; for a gradient

m

, the perpendicular gradient is

-\frac{1}{m}

odd function

A function with rotational symmetry about the origin, satisfying

f\left(-x\right)=-f\left(x\right)

, for example

\sin\left(-x\right)=-\sin x

and

\tan\left(-x\right)=-\tan x

opposite

The side that lies across from the chosen acute angle in a right-angled triangle.

parallel

Checks consistency by comparing results from two different versions designed to measure the same thing.

parametric form

A way to describe a line by writing each coordinate as a function of a parameter, for example $x=x_0+λl$ , $y=y_0+λm$ , $z=z_0+λn$ .

perpendicular

Meeting at a right angle, so the angle between the two lines is

90^{\circ}

perpendicular bisector

A line that passes through the midpoint of a line segment and is perpendicular to the segment.

point-gradient

A form of a straight-line equation using a point

(x_1,y_1)

and gradient

m

y-y_1=m\left(x-x_1\right)

position

The vector from the origin $O$ to a point, written for example as $\rightarrow{OA}=\textbf{a}$ .

position vector

A vector from the origin to a point, used to locate the point in space; in $r=a+\lambda b$ , $\textbf{a}$ is the position vector of a fixed point on the line.

Prim’s algorithm

Finds a minimum spanning tree by starting at a vertex and repeatedly adding the smallest edge that connects the current tree to a new vertex, until all vertices are included.

probabilities

Numbers between

0

and

1

that measure how likely a move or outcome is, with totals summing to

1

across all possible next states under the stated convention.

pyramid

A solid with a polygon base and triangular faces that meet at a single vertex; for a right pyramid, $V=\frac{1}{3}Ah$ where $A$ is the base area and $h$ is the perpendicular height.

quadrant

One of the four regions of the coordinate plane determined by the

x

- and

y

-axes, used to determine the signs of

\sin\theta

\cos\theta

, and

\tan\theta

radian

A unit for measuring angles defined so that $\theta$ radians equals the arc length divided by the radius, $\theta=\frac{l}{r}$ , giving the key conversion $\pi$ radians $=180^\circ$ .

radius

The distance from the centre of a circle to any point on the circle.

reference angle

The acute angle between the terminal side of an angle and the

x

-axis, used with quadrant signs to find exact trigonometric values.

reflection

A transformation that flips points in a line (such as an axis), reversing orientation while preserving distances.

Reverse integration

Describes using the chain rule in reverse to integrate expressions matching the pattern '

g'(x)f\left(g(x)\right)

', giving '

F\left(g(x)\right)+C

' where '

F'(u)=f(u)

right pyramid

A pyramid whose apex is vertically above the centre of its base; its volume is $V=\frac{1}{3}Ah$ , where $A$ is the base area and $h$ is the perpendicular height.

rotation

A transformation about the origin through angle $\theta$ anticlockwise, represented by $\begin{pmatrix}\cos\theta&&-\sin\theta\\\sin\theta&&\cos\theta\end{pmatrix}$

scalar

A single number used to multiply a matrix by multiplying every element by that number.

scalar parameter

A real number (such as $λ$ or $μ$ ) that varies to generate different points on a line in a parametric or vector equation.

sector

The region enclosed by two radii and the arc between them.

simple

Describes a graph with no loops and no repeated edges between the same pair of vertices.

sin

A trigonometric function whose derivative is '

\cos x

sine

A trigonometric ratio/function that can be expanded for compound angles using $\\sin\left(A\pm B\right)=\sin A\cos B\pm\cos A\sin B$ .

sine rule

A relationship for any triangle with sides

a

b

c

opposite angles

A

B

C

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

skew

Describes two lines in three dimensions that are not parallel and do not meet, so no common solution exists when their parametric equations are solved together.

sphere

A solid consisting of all points at a fixed distance $r$ from a centre; its volume is $V=\frac{4}{3}\pi r^3$ .

subgraph

A smaller graph formed using some of the vertices and some of the edges of a larger graph, where the chosen edges join only chosen vertices.

tan

Defined by

\tan\theta=\frac{\sin\theta}{\cos\theta}

, so it is undefined whenever

\cos\theta=0

tangent

A line that touches a curve at a point and has the same local slope as the curve there, used to interpret the derivative at that point.

transformation

A change to a geometric object that can alter its position, orientation, or size.

translation vector

A vector $\begin{pmatrix}e\\f\end{pmatrix}$ that shifts every point by the same amount, giving $\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}e\\f\end{pmatrix}$

tree

A connected graph with no cycles; if it has

n

vertices then it has exactly

n-1

edges.

unit circle

A circle of radius

1

centred at the origin, with equation

x^2+y^2=1

, used to define trigonometric functions for any angle.

vector

An operation on two vectors that returns a vector perpendicular to both, with direction given by the right-hand rule and magnitude $|\textbf{v}||\textbf{w}|\text{sin}\theta$ .

vector equation

A representation of a line in 2D or 3D written as a position vector $\textbf{r}$ in terms of a fixed point and a scalar parameter, typically in the form $r = a + \lambda b$ .

vertex

A point in a graph.

vertical

Describing changes parallel to the

y

-axis, such as a stretch by factor

|a|

or a shift up or down by

d

vertical stretch

A transformation that multiplies the $y$ -coordinate by a scale factor $k$ while leaving the $x$ -coordinate unchanged, using $\begin{pmatrix}1&&0\\0&&k\end{pmatrix}$

vertices

Points where three edges meet, so the location is the same distance from three sites.

Volume

A measure of how much space a three-dimensional solid occupies, usually expressed in cubic units such as cm³.

Voronoi

A diagram that partitions the plane into regions based on distance to a given set of points, so each region contains the points closest to one particular point.

walks

Routes through a graph formed by moving along edges from one vertex to another, where vertices and edges may be repeated.

weighted

Describes a graph whose edges have numbers attached to them to represent quantities such as distance, cost, or time.

weights

Numerical values attached to edges to represent quantities such as distance, cost, or time.

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