International Baccalaureate Mathematics

Syllabus Content

Concept of a vector; position vectors; displacement vectors.
Representation of vectors using directed line segments.
Base vectors i, j, k.
Components of a vector:
v=v₁i+v₂j+v₃k.
Algebraic and geometric approaches to the following:
the sum and difference of two vectors
the zero vector 0, the vector -v
multiplication by a scalar, kv, parallel vectors
magnitude of a vector, |v|; unit vectors, v÷|v|
position vectors OA=a, OB=b
displacement vector AB=b-a
Proofs of geometrical properties using vectors.

Here are some exam-style questions on this statement:

"The line L is parallel to the vector $\begin{pmatrix} 2 \\ 5 \\ \end{pmatrix} $," ... more

"The following diagram shows parallelogram OABC with $ \overrightarrow{OA}\ = \pmb{a} $, $ \overrightarrow{OC}\ = \pmb{c} $ and $|\pmb{c}| = 2|\pmb{a}| $" ... more

"Two points $A$ and $B$ have coordinates (1 , 3 , 6) and (8 , 7 , 10) respectively." ... more

"The points A and B have coordinates $(3,-2,1)$ and $(4, 0, -1)$ respectively." ... more

"Consider the vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\mathbf{a} = \begin{pmatrix} 16 \\ -12 \end{pmatrix} $ and $ |\mathbf{b}| = 11$." ... more

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Vectors Vectors usually first make an appearance when pupils learn about transformations. A translation is best described with a vector written as a two by one matrix. Following that pupils learn how vectors can be used to prove geometric relationships in simple line diagrams. Ultimately vectors are studied as a major topic for A Level and International Baccalaureate courses where vectors in three dimensions are included.You can think of a vector as what is needed to 'carry' the point A to the point B. The Latin word vector means ‘carrier’ and was first used by 18th century astronomers investigating planet rotation around the Sun.

Furthermore

Official Guidance, clarification and syllabus links:

Distance between points $A$ and $B$ is the magnitude of $ \overrightarrow{AB} $

Formula Booklet for 3.12:

Magnitude of a vector

$ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \quad \text{, where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} $

In the field of mathematics, particularly in vector algebra, a vector is a fundamental concept that represents both magnitude and direction. Vectors can be added or subtracted to form new vectors, and they can also be scaled (multiplied) by a scalar, which changes their magnitude without altering their direction. The magnitude of a vector is a measure of its length.

The key formulae for vectors are:

Vector Addition: $\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}$

Vector Subtraction: $\mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}$

Scalar Multiplication: $k\mathbf{a} = k \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} ka_1 \\ ka_2 \end{pmatrix}$

Magnitude of a Vector: $\|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2}$

Example:
Consider two vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$.
The sum of $\mathbf{a}$ and $\mathbf{b}$ is: $$\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$$
If we multiply vector $\mathbf{a}$ by a scalar, say 3, we get: $$3\mathbf{a} = 3 \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times 3 \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$$
The magnitude of vector $\mathbf{a}$ is: $$\|\mathbf{a}\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$$

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International Baccalaureate Mathematics

Geometry and Trigonometry

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