Grade 11 Advanced Mathematics - Term 3 Vocabulary

Comprehensive vocabulary flashcards focused on Grade 11 Advanced Mathematics Term 3 topics including vector operations, polar graphs, sequences, series, and complex numbers.

Component Form of a Vector

The representation of a vector as $\langle x, y \rangle$, calculated by subtracting the initial point coordinates from the terminal point coordinates: $\langle x_2 - x_1, y_2 - y_1 \rangle$.

Dot Product

An operation on two vectors $\mathbf{u}$ and $\mathbf{v}$ defined as $\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z$. If the result is $0$, the vectors are orthogonal.

Orthogonal Vectors

Two vectors whose dot product is exactly equal to $0$, meaning they are perpendicular to each other.

Cross Product

A vector operation in three-dimensional space used to find the area of a parallelogram (using $A = |u \times v|$) or the volume of a parallelopiped.

Polar Coordinate System

A system that locates a point $(r, \theta)$ based on its distance $r$ from the pole and its angle $\theta$ from the polar axis.

Polar Distance Formula

The formula used to find the distance between two points in polar coordinates: $d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2 \cos(\theta_2 - \theta_1)}$.

Rose Curve

A classical polar curve defined by $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$. It has $n$ petals if $n$ is odd and $2n$ petals if $n$ is even.

Limacons

A type of polar graph represented by equations of the form $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$. If $a = b$, the graph is specifically called a cardioid.

Lemniscates

A polar graph shaped like a figure eight or propellor, represented by $r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$.

Spirals of Archimedes

A polar graph defined by the equation $r = a\theta + b$.

Absolute Value of a Complex Number

The distance of a complex number $z = a + bi$ from the origin in the complex plane, calculated as $|z| = \sqrt{a^2 + b^2}$.

Arithmetic Sequence

A sequence of numbers in which the difference between consecutive terms is constant; it is mathematically related to linear functions.

Geometric Sequence

A sequence of numbers where each term after the first is found by multiplying the previous one by a constant ratio $r$; it is mathematically related to exponential functions.

Sigma Notation

The symbol $\sum$ used to represent the sum of a series, often indicating a lower bound and an upper bound for the index.

Convergent Sequence

A sequence that has a limit such that the terms approach a unique number as the index increases.

Divergent Sequence

A sequence that does not approach a unique number as its terms progress.

Geometric Means

The terms between two nonconsecutive terms of a geometric sequence.

Recursive Formula

A formula that defines a term $a_n$ of a sequence in relation to one or more preceding terms, such as $a_{n-1}$.

Iterates

A sequence of values $(x_1, x_2, x_3, ...)$ produced by repeatedly applying a function to an initial value $x_0$.

Binomial Theorem

A theorem used to expand powers of binomials, typically taking the form $(a + b)^n$.

Counter-example

A specific case or value that proves a general mathematical statement or conjecture is false.

Prime Number

A whole number greater than $1$ whose only divisors are $1$ and itself.