IB AA: Vectors in 2D and 3D

Vectors are the mathematical language of direction and magnitude, providing a powerful framework for modeling everything from physics forces to computer graphics. In the IB Analysis & Approaches (AA) course, mastering vectors in two and three dimensions is essential, as they form a critical bridge between algebraic techniques and geometric intuition, with deep applications in proof and problem-solving.

Vector Fundamentals: Notation and Components

A vector is a quantity possessing both magnitude (size) and direction, contrasting with a scalar, which has only magnitude. Vectors are typically represented in bold, like $\vec{v}$, or as a column matrix. In a coordinate system, a 2D vector is defined by its horizontal ($x$) and vertical ($y$) components, written as $\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ or $\vec{v} = x\vec{i} + y\vec{j}$, where $\vec{i}$ and $\vec{j}$ are the standard unit vectors along the $x$ and $y$-axes. In 3D, a third component ($z$) and unit vector $\vec{k}$ are added: $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = x\vec{i} + y\vec{j} + z\vec{k}$.

The magnitude (or length) of a vector $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is found using a generalized Pythagorean theorem: $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$. A position vector is a special vector that describes the location of a point relative to a fixed origin, $O$. If point $A$ has coordinates $(a_1, a_2, a_3)$, its position vector is $\vec{OA} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$. The vector connecting two points, $\vec{AB}$, is found by subtracting their position vectors: $\vec{AB} = \vec{OB} - \vec{OA}$.

Vector Operations: Addition and Scalar Multiplication

These two core operations form the algebra of vectors. Vector addition is performed component-wise: if $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$, then $\vec{a} + \vec{b} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \end{pmatrix}$. Geometrically, this corresponds to placing the tail of $\vec{b}$ at the head of $\vec{a}$; the resultant vector is the direct path from the tail of $\vec{a}$ to the head of $\vec{b}$ (the triangle or parallelogram rule).

Scalar multiplication involves multiplying a vector by a real number (a scalar) $k$. This is also done component-wise: $k\vec{a} = \begin{pmatrix} ka_1 \\ ka_2 \\ ka_3 \end{pmatrix}$. Geometrically, it scales the vector's length by a factor of $|k|$. If $k$ is positive, the direction is unchanged; if $k$ is negative, the direction is reversed. A unit vector in the direction of $\vec{a}$ is found via $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$.

The Scalar (Dot) Product and Angles

The scalar product or dot product of two vectors yields a scalar result. It is defined algebraically as $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$. Its true power lies in its geometric meaning, which connects algebra to geometry: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$, where $\theta$ is the angle between vectors $0^\circ \leq \theta \leq 180^\circ$.

This formula allows you to calculate the angle between any two non-zero vectors: $$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}.$$ If $\vec{a} \cdot \vec{b} = 0$ and neither vector is zero, the vectors are perpendicular (orthogonal). The dot product also gives the scalar projection of $\vec{a}$ onto $\vec{b}$: $\text{comp}_{\vec{b}}\ \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$, which represents the length of the "shadow" of $\vec{a}$ in the direction of $\vec{b}$.

The Vector (Cross) Product - Higher Level

For HL students, the vector product or cross product is a crucial extension into 3D. Defined only for three-dimensional vectors, it takes two vectors and returns a vector. The algebraic definition uses a determinant form: $$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\vec{i} - (a_1b_3 - a_3b_1)\vec{j} + (a_1b_2 - a_2b_1)\vec{k}.$$

Geometrically, the resulting vector $\vec{c} = \vec{a} \times \vec{b}$ has two key properties. First, it is perpendicular to both $\vec{a}$ and $\vec{b}$ (its direction is given by the right-hand rule). Second, its magnitude equals the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$: $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$. This makes it invaluable for calculating areas and finding normal vectors to planes.

Applications to Geometry Proofs

Vectors provide an elegant, coordinate-based method for proving geometric theorems. Common strategies include:

• Proving collinearity: Show that the vector between points is a scalar multiple of another. If $\vec{AB} = k \vec{BC}$, then points $A$, $B$, and $C$ lie on the same line.
• Proving parallelism: Demonstrate that one vector is a scalar multiple of another.
• Finding midpoints: The position vector of the midpoint $M$ of $AB$ is $\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB})$.
• Proving perpendicularity: Show that the dot product of two direction vectors is zero.
• Calculating areas: Use the magnitude of the cross product to find the area of a triangle ($\frac{1}{2} |\vec{AB} \times \vec{AC}|$) or parallelogram.

For example, to prove the diagonals of a parallelogram bisect each other, you could assign position vectors to the vertices and show the position vectors of the midpoints of both diagonals are identical.

Common Pitfalls

1. Confusing Points and Vectors: A point is a location; a vector is a displacement. The position vector $\vec{OA}$ is the displacement from the origin $O$ to point $A$. The vector $\vec{AB}$ is the displacement from $A$ to $B$. Always be clear whether you are working with a point's coordinates or a vector's components.
2. Misapplying the Dot and Cross Product: The dot product yields a scalar and is commutative ($\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$). The cross product yields a vector and is anti-commutative ($\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}$). Using the wrong product when finding an angle or checking perpendicularity is a critical error. Remember: dot product for angles and perpendicularity in any dimension; cross product for perpendicular vectors and areas in 3D only.
3. Neglecting Direction in Scalar Multiplication: When finding a unit vector, you must divide each component by the magnitude. Simply dividing one component is incorrect. Similarly, when a vector is multiplied by a negative scalar, its direction reverses completely.
4. Arithmetic Errors in 3D: With three components, calculations become longer. A single sign error in the dot product will corrupt your angle calculation, and a minor arithmetic mistake in the cross product determinant will yield an incorrect perpendicular vector. Work methodically and check your steps.

Summary

• Vectors are defined by magnitude and direction, represented via components or unit vector notation. The position vector $\vec{OA}$ locates point $A$ from the origin.
• Core operations are component-wise addition and scalar multiplication. The dot product $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ finds angles and tests for perpendicularity.
• The cross product $\vec{a} \times \vec{b}$ (HL) produces a vector perpendicular to both $\vec{a}$ and $\vec{b}$, with magnitude equal to the area of the spanned parallelogram.
• Vectors are powerful tools for geometric proofs, enabling elegant demonstrations of collinearity, parallelism, and bisection through algebraic manipulation of position and direction vectors.
• Success requires meticulous attention to the distinctions between points and vectors, scalars and vectors, and the specific geometric interpretations of the dot and cross products.