Vectors: Complete Lesson (coordinates, collinearity, dot product)

Definition

A vector $\vec{AB}$ is determined by three things:

• A direction (that of line $(AB)$)
• A sense (from $A$ to $B$)
• A magnitude (the length $AB$)

Two vectors with same direction, sense and magnitude are equal, even if they start from different points.

Coordinates in a frame

In a frame $(O, \vec{i}, \vec{j})$, if $A(x_A, y_A)$ and $B(x_B, y_B)$:

$\vec{AB} \begin{pmatrix} x_B - x_A \\ y_B - y_A \end{pmatrix}$

Watch out: “endpoint minus start”, not the other way.

Operations

Addition (Chasles)

$\vec{AB} + \vec{BC} = \vec{AC}$

In coordinates: add component-wise.

Scalar multiplication

If $k \in \mathbb{R}$ and $\vec{u} = (x, y)$: $k \vec{u} = (kx, ky)$

• $k > 0$: same sense, norm multiplied by $|k|$.
• $k < 0$: opposite sense.
• $k = 0$: zero vector.

Norm of a vector

$\|\vec{u}\| = \sqrt{x^2 + y^2}$

The vector’s length.

Example: $\vec{u}(3, 4)$ ⇒ $\|\vec{u}\| = 5$.

Collinearity

Vectors $\vec{u}(x_1, y_1)$ and $\vec{v}(x_2, y_2)$ are collinear iff:

$\det(\vec{u}, \vec{v}) = x_1 y_2 - x_2 y_1 = 0$

Uses:

• 3 points $A, B, C$ collinear ⇔ $\vec{AB}, \vec{AC}$ collinear.
• Parallel lines ⇔ collinear direction vectors.

Dot product

$\vec{u} \cdot \vec{v} = x_u x_v + y_u y_v$

Also: $\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos(\theta)$

Key property (orthogonality)

$\vec{u} \perp \vec{v} \iff \vec{u} \cdot \vec{v} = 0$

The perpendicularity test.

Midpoint

If $I$ is the midpoint of $[AB]$: $I\left(\dfrac{x_A + x_B}{2}, \dfrac{y_A + y_B}{2}\right)$

Vectorially: $\vec{AI} = \vec{IB}$.

Sample exercise

$A(1, 2)$, $B(4, 6)$, $C(7, 10)$.

Q1: Are they collinear?

$\vec{AB} = (3, 4)$, $\vec{AC} = (6, 8)$. $\det = 3 \times 8 - 6 \times 4 = 0$. Yes.

Q2: Distance $AB$.

$\|\vec{AB}\| = \sqrt{25} = 5$.

Q3: Midpoint $M$ of $[AC]$.

$M(4, 6)$. Notice: $M = B$! (points are collinear).

Common mistakes

1. Confusing point and vector coordinates: $\vec{AB} = (x_B - x_A, y_B - y_A)$.
2. Using slope to test collinearity: determinant is safer (no division by zero).
3. Adding norms: $\|\vec{u} + \vec{v}\| \neq \|\vec{u}\| + \|\vec{v}\|$ generally.

Takeaways

• Vector coordinates: endpoint − start.
• $\|\vec{u}\| = \sqrt{x^2 + y^2}$.
• Collinear ⇔ $\det = 0$.
• Orthogonal ⇔ dot product = 0.
• Chasles: $\vec{AB} + \vec{BC} = \vec{AC}$.
• Midpoint: average of coordinates.

Going further

• Related tips: Collinear vectors (determinant), Dot product (3 formulas).
• Next: dot product in 3D.