3.11 Vectors: Algebraic Representation (HL)

1. 2-Dimensional Algebraic Vectors

A vector $\vec{u}$ is formally expressed algebraically as a column matrix containing its directional components: $\binom{a}{b}$.

Magnitude: The absolute length of the vector evaluates utilizing Pythagorean distance: $|\vec{u}| = \sqrt{a^2 + b^2}$.

Addition: Component rows are summed independently. $\binom{a_1}{b_1} + \binom{a_2}{b_2} = \binom{a_1 + a_2}{b_1 + b_2}$.

Scalar Multiplication: The scalar multiplier distributes across all components. $k\binom{a}{b} = \binom{ka}{kb}$. The resulting magnitude scales proportionally: $|k\vec{u}| = |k| \cdot |\vec{u}|$.

EXAMPLE 1 & 2 (Algebraic Operations)

Given vectors $\vec{u} = \binom{3}{4}$ and $\vec{v} = \binom{2}{5}$:

Magnitudes: $|\vec{u}| = \sqrt{3^2 + 4^2} = \mathbf{5}$. $|\vec{v}| = \sqrt{2^2 + 5^2} = \mathbf{\sqrt{29}}$.

Addition: $\vec{u} + \vec{v} = \binom{3+2}{4+5} = \mathbf{\binom{5}{9}}$.

Linear combinations: $2\vec{u} + 3\vec{v} = \binom{6}{8} + \binom{6}{15} = \mathbf{\binom{12}{23}}$.

2. Unit Vectors and Base Notation

The Unit Vector: A unit vector $\hat{u}$ is a specialized vector possessing a magnitude of exactly 1, pointing precisely in the same direction as $\vec{u}$. It is calculated by dividing the vector by its own magnitude:

$\hat{u} = \frac{1}{|\vec{u}|} \vec{u}$

EXAMPLE 3

Establish a vector $\vec{a}$ parallel to $\vec{u} = \binom{3}{4}$ possessing a strict magnitude of 20.

The magnitude of $\vec{u}$ is 5. Because the target magnitude is exactly 4 times larger ($20/5 = 4$), the solution evaluates to $\vec{a} = 4\vec{u} = \mathbf{\binom{12}{16}}$.

General Methodology: Formulate the unit vector first ($\hat{u} = \binom{3/5}{4/5}$), then multiply by the targeted length $20$.

Standard Base Vectors ($\vec{i}$ and $\vec{j}$):

Any 2D vector can be expressed as a linear combination of the fundamental unit vectors tracing the x-axis and y-axis. Defining $\vec{i} = \binom{1}{0}$ and $\vec{j} = \binom{0}{1}$, the column vector $\binom{a}{b}$ translates algebraically to $a\vec{i} + b\vec{j}$.

3. 3-Dimensional Vectors and Point Mapping

Expanding into 3D space introduces a third vertical $z$-axis component. Vectors formulate as $\binom{a}{b}{c}$ or algebraically as $a\vec{i} + b\vec{j} + c\vec{k}$. The magnitude scales identically: $|\vec{u}| = \sqrt{a^2 + b^2 + c^2}$.

EXAMPLE 4

Evaluate operations for $\vec{u} = \binom{1}{2}{3}$ and $\vec{v} = \binom{2}{5}{-4}$.

$3\vec{u} + 2\vec{v} = \binom{3}{6}{9} + \binom{4}{10}{-8} = \mathbf{\binom{7}{16}{1}}$.

Unit vector $\hat{u} = \frac{1}{\sqrt{14}}\binom{1}{2}{3} = \mathbf{\binom{1/\sqrt{14}}{2/\sqrt{14}}{3/\sqrt{14}}}$.

Vector Mapping Between Points

The directional vector spanning from point $A(x_1, y_1, z_1)$ to point $B(x_2, y_2, z_2)$ is mathematically extracted by subtracting the position vector of the origin tail ($A$) from the position vector of the destination head ($B$):

$\vec{AB} = \vec{OB} - \vec{OA} = \binom{x_2 - x_1}{y_2 - y_1}{z_2 - z_1}$