3.10 Vectors: Geometric Representation (HL)

1. Scalars versus Vectors

Physical and mathematical quantities are strictly classified into two distinct operational categories:

Scalars: Possess only magnitude (size). Examples include age ($28$), length ($4\text{m}$), and temperature ($25^\circ\text{C}$).

Vectors: Possess both magnitude and a specific directional orientation. Examples include force ($7\text{N}$ downwards) and velocity ($35\text{m/s}$ towards southeast).

Geometrically, a vector is represented as a directed arrow. It is denoted symbolically by a letter ($\vec{u}$) or by its terminal endpoints ($\vec{AB}$, where $A$ is the tail and $B$ is the head). The absolute length of the vector defines its magnitude, denoted as $|\vec{u}|$ or $|\vec{AB}|$.

Equality of Vectors

Two vectors are classified as completely equal ($\vec{u} = \vec{v}$) if and only if they possess identical magnitudes and identical directions. A vector is not bound to a fixed spatial coordinate; it remains mathematically identical regardless of parallel translation.

In a standard parallelogram $ABCD$, the parallel boundary lines establish vector equivalencies: $\vec{AB} = \vec{DC}$, $\vec{AD} = \vec{BC}$, $\vec{BA} = \vec{CD}$, and $\vec{DA} = \vec{CB}$.

2. Geometric Vector Operations

Addition ($\vec{u} + \vec{v}$): Executed physically by placing vectors "head-to-tail". Moving from point $A$ to $B$, and sequentially from $B$ to $C$, produces the resultant vector traversing directly from $A$ to $C$. Thus, $\vec{AB} + \vec{BC} = \vec{AC}$.

The Opposite Vector ($-\vec{u}$): Maintains the exact identical magnitude but reverses the directional orientation by $180^\circ$. Using endpoint notation: $\vec{AB} = -\vec{BA}$.

The Zero Vector ($\vec{0}$): An entity possessing exactly zero magnitude and lacking any defined direction. Arises when a vector cancels itself: $\vec{AB} - \vec{AB} = \vec{AA} = \vec{0}$.

Scalar Multiplication ($k\vec{u}$): For any positive scalar $k > 0$, the product $k\vec{u}$ generates a parallel vector scaling the magnitude by $k$. If $k < 0$, the magnitude scales by $|k|$ while the direction strictly reverses. Two vectors are defined as parallel ($\vec{u} \parallel \vec{v}$) if $\vec{u} = k\vec{v}$ for some scalar $k \in \mathbb{R}$.

EXAMPLE 1 (Vector Pathways in a Cube)

Consider a standard cube $ABCDEFGH$. Base vectors are assigned to edges: $\vec{a} = \vec{AB}$, $\vec{b} = \vec{AD}$, and $\vec{c} = \vec{AF}$.

Parallel edges equate directly to these base vectors: $\vec{a} = \vec{DC} = \vec{FG} = \vec{EH}$.

To express the diagonal vector $\vec{FC}$ exclusively in terms of $\vec{a}, \vec{b}, \vec{c}$, a continuous path must be mapped from node $F$ to node $C$.

Path formulation: $\vec{FC} = \vec{FG} + \vec{GB} + \vec{BC}$.

Substituting the equivalent base vectors yields: $\vec{FC} = \vec{a} - \vec{c} + \vec{b}$.

EXAMPLE 2 (Cartesian Translation)

Any vector floating on a 2D Cartesian plane can undergo a parallel translation to originate precisely from the origin $O(0,0)$.

If a vector's head terminates at coordinate $A(a,b)$ after originating from $O$, it is strictly designated as the position vector of point $A$, denoted as $\vec{OA} = \binom{a}{b}$. For instance, a vector translating 3 units horizontally and 4 units vertically maps to position vector $\binom{3}{4}$.