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Section 1.2: Vector Fundamentals
Understand what vectors are, how they represent physical quantities, and how to break them down into their components and different notations.

Section Introduction

To be able to study motion, forces, and velocities, one must develop an understanding of quantities with both size and direction. This is where vectors come in. In this section, we'll define what a vector is, distinguish it from a scalar, explore how we represent vectors using components in 1D, 2D, and 3D space, and learn how to calculate a vector's magnitude (length).

Surya: Okay, I get coordinates – like (3, 2, 5). But how do I tell a robot arm to *move*? Knowing the start and end isn't enough, right?

Leo: Exactly! That's where come in. Think of kicking a ball: you need both *how hard* (magnitude) and *in what direction*. A vector captures both.

Surya: So, coordinates = location, vectors = movement, force, or velocity (speed + direction)?

Leo: You got it! Coordinates are map pins. Vectors are arrows showing the journey or force.

What are Vectors? Scalar vs. Vector

Sometimes, location isn't the whole story. We often need quantities with both size () and direction.

:
Has magnitude only (e.g., temperature: 25°C).
:
Has magnitude and direction (e.g., pushing a box: strength + direction).

Examples of vectors include , (speed in a specific direction), and (change in position).

Vectors in One Dimension (1D)

Imagine a number line. A 1D vector simply represents movement or quantity along that line.

  • : The absolute value of the number (e.g., |+5| = 5, |-3| = 3).
  • : Indicated by the sign (+ for positive direction, - for negative).

Example: A vector +6+6 means moving 6 units in the positive direction.

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Extending to Two Dimensions (2D)

In 2D (like a flat plane), we need two pieces of information to describe a vector's direction and magnitude, typically using the X and Y axes. We break the vector down into :

  • VxV_x: How far the vector extends along the X-axis.
  • VyV_y: How far the vector extends along the Y-axis.

The magnitude (length) in 2D is found using the standard Pythagorean theorem:

Key Formulas

∥V⃗∥=Vx2+Vy2\|\vec{V}\| = \sqrt{V_x^2 + V_y^2}

This calculates the hypotenuse of a right triangle formed by the vector's components.

Visually, it's an arrow on a 2D coordinate plane:

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Anatomy of a Vector (3D)

Now that we've seen how vectors work in 1D and 2D, let's add the third dimension (Z-axis) to fully understand vectors in 3D space.

In 3D space, a vector is still represented by an arrow. Its Tail marks the starting point, and the Head (tip) marks the ending point. The arrow's Length represents the vector's Magnitude (size, strength, or speed), and its Orientation in space indicates its Direction.

Changing the arrow's start or end points alters its length and direction. This visual helps, but for precision, we need numbers: .

Describing 3D Vectors Precisely: Components & Notation

To work with 3D vectors mathematically, we break them into along the X, Y, and Z axes. These components (VxV_x, VyV_y, VzV_z) tell us how much the vector extends along each axis.

Key Formulas

To calculate the components of a 3D vector with tail at P1(x1,y1,z1)P_1(x_1, y_1, z_1) and head at P2(x2,y2,z2)P_2(x_2, y_2, z_2):

    Change in X:

  • Vx=x2−x1=ΔxV_x = x_2 - x_1 = \Delta x

    • Change in Y:

  • Vy=y2−y1=ΔyV_y = y_2 - y_1 = \Delta y

    • Change in Z:

  • Vz=z2−z1=ΔzV_z = z_2 - z_1 = \Delta z

You may also encounter vectors written like this:

    Angle Bracket Notation:

  • V⃗=⟨Vx,Vy,Vz⟩\vec{V} = \langle V_x, V_y, V_z \rangle

    • Parentheses Notation:

  • V⃗=(Vx,Vy,Vz)\vec{V} = (V_x, V_y, V_z)

Vector Magnitude in 3D: Measuring Length

The (length) of a 3D vector V⃗\vec{V} with components VxV_x, VyV_y, VzV_z extends the 2D concept using the 3D Pythagorean theorem:

Key Formulas
Where:
  • ∥V⃗∥\|\vec{V}\|: Represents the magnitude (or length) of the vector V⃗\vec{V}.
  • VxV_x: The component of the vector along the X-axis.
  • VyV_y: The component of the vector along the Y-axis.
  • VzV_z: The component of the vector along the Z-axis.

∥V⃗∥=Vx2+Vy2+Vz2\|\vec{V}\| = \sqrt{V_x^2 + V_y^2 + V_z^2}

This formula calculates the vector's overall length by combining its individual extensions along the X, Y, and Z axes. Think of it as finding the diagonal length of a 3D box defined by the vector's components.

Hands-On: Vector Builder

Adjust the start and end points of the vector using the controls. Observe how the vector's components (Vx, Vy, Vz) and its angle bracket notation change in real-time.

Vector Properties

ΔX: 0.00

ΔY: 0.00

ΔZ: 0.00

Vector: --

Magnitude: 0.00

Define Vector

Start Point (X1, Y1, Z1)
X
Y
Z
End Point (X2, Y2, Z2)
X
Y
Z

Section Review: Vectors and Components

Check Your Understanding

A vector pointing directly upwards has:

If you double the magnitude of a vector, what happens to its length?