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Lesson Directive // Vector MathematicsREF_CORE

What is a Vector?

v⃗\vec{v}2^2==vxv_x2^2++vyv_y2^2

Hover over a variable in the formula above, or see glossary below:

v⃗\vec{v}
Total Velocity
m/s
vxv_x
X-Velocity
m/s
vyv_y
Y-Velocity
m/s

Imagine drawing an arrow on a map. A vector is exactly that: a mathematical arrow that has both a length (called magnitude) and a direction. Unlike a simple number like '5 kg' (a scalar), a vector tells you '5 m/s going North'.

INSIGHT: A vector requires two pieces of information: how much (magnitude), and which way (direction).

Vector Components

Instead of dealing with angles, it's often easier to break that diagonal arrow into a horizontal step (X) and a vertical step (Y). Any diagonal movement is just a combination of moving sideways and moving up or down.

INSIGHT: You can perfectly recreate any diagonal path by moving purely horizontally then vertically.

The Pythagorean Theorem

Because the X and Y steps form a 90-degree right angle, we can use the Pythagorean theorem (a^2 + b^2 = c^2) to find the length of the hypotenuse, which gives us the original vector's total length (magnitude).

INSIGHT: Squaring the components, adding them, and taking the square root gives you the total speed.

Vector Multiplication (Dot Product)

What happens when you multiply two vectors? The Dot Product multiplies the parts of two vectors that are pointing in the same direction. It outputs a single normal number (a scalar). In physics, pushing a spacecraft (Force vector) in the direction it's moving (Displacement vector) gives us Work (Energy).

INSIGHT: Dot Product = How much do these two arrows align with each other?

Vector Multiplication (Cross Product)

The Cross Product is the opposite: it multiplies the parts of two vectors that are perpendicular to each other, and it creates a brand new 3D vector pointing straight out of the surface (like a screw turning). This is used to calculate orbital Torque and Angular Momentum.

INSIGHT: Cross Product = How perpendicular are these two arrows, and what is the resulting 3D twist?

Transformations and Thrust

When a spacecraft fires its thrusters, it essentially 'adds' a new thrust vector to its current velocity vector. If you want to change direction, you calculate a transformation to figure out exactly which angle to point your engines.

INSIGHT: Thrusting forward adds to your magnitude (speed). Thrusting sideways changes your direction.
Detailed Theory & ReferencesEXT_DOC

Vector Algebra and State Vectors

A vector is a mathematical object that has both a magnitude (length) and a direction. In classical mechanics, physical quantities like position, velocity, and force must be described as vectors, as opposed to scalars (which have only magnitude, like mass or temperature).

Vector Notation and Components

In three-dimensional Cartesian space, a vector v⃗\vec{v} can be represented by its orthogonal components along the x, y, and z axes: v⃗=[xyz]=xi^+yj^+zk^\vec{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} = x\hat{i} + y\hat{j} + z\hat{k}

The magnitude of the vector is given by the Euclidean norm: ∣v⃗∣=x2+y2+z2|\vec{v}| = \sqrt{x^2 + y^2 + z^2}

The Orbital State Vector

In astrodynamics, the orbital state vector is the absolute foundation of spacecraft navigation. It consists of two 3-dimensional vectors grouped together:

  1. Position Vector (r⃗\vec{r}): Determines the exact 3D coordinate of the spacecraft relative to the central body.
  2. Velocity Vector (v⃗\vec{v}): Determines the 3D speed and heading of the spacecraft.

Combined, these six variables [x,y,z,vx,vy,vz][x, y, z, v_x, v_y, v_z] completely and deterministically define the spacecraft's current orbit. From a single accurate state vector reading at a known time tt, Kepler's laws can be mathematically applied to project the spacecraft's exact position infinitely into the future (assuming no perturbing forces).

Reference: Anton, H., & Rorres, C. (2013). Elementary Linear Algebra (11th ed.). Wiley.

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.