Lesson 4.1.2

Translations via Vectors

A translation slides every point of a figure the same distance in the same direction. The instruction set for a translation is a vector — an arrow with a specific length and direction.

Introduction

You already know that a translation is a “slide.” In this lesson you'll learn to describe that slide precisely using a vector — the mathematical way to encode both direction and magnitude in one neat package.

Past Knowledge

Pre-image / image vocabulary (4.1.1). Coordinate plane basics (1.2).

Today's Goal

Define vectors, use component form, and translate figures using vector notation.

Future Success

Coordinate translations (4.1.3) and compositions (4.2.2) build directly on vector notation.

Key Concepts

What Is a Vector?

A vector is a quantity that has both magnitude (length) and direction. We draw it as an arrow from an initial point to a terminal point.

Component Form

= horizontal change (positive → right), = vertical change (positive → up).

Translation Rule (Vector Form)

Every point in the pre-image is shifted right units and up units.

Key Properties of Translations

  • Rigid — distances and angles are preserved.
  • Orientation preserved — the figure does not flip.
  • Every segment in the image is parallel to the corresponding pre-image segment and equal in length.

Worked Examples

Basic

Writing a Vector from Two Points

A point moves from to . Write the translation vector.

— right 5, down 2.

Vector
⟨,⟩
Adjust the vector to explore
P(2,3)→P′(7,1)
Intermediate

Translating a Triangle

Translate with , , by .

Vector
⟨,⟩
Adjust the vector to explore
A(0,4)→A′(-3,9)
B(3,1)→B′(0,6)
C(-2,-1)→C′(-5,4)
Advanced

Finding the Vector from Pre-image to Image

has vertices , , . After a translation, . Find the vector and the remaining image vertices.

From to :

Apply to the other vertices:

;

Vector
⟨,⟩
Adjust the vector to explore
D(1,2)→D′(-2,5)
E(4,6)→E′(1,9)
F(5,0)→F′(2,3)

Common Pitfalls

Confusing “Negative” Direction

means left 3 and up 5 — not right 3. Pay careful attention to the sign of each component.

Subtracting in the Wrong Order

To find the vector, always subtract: (image minus pre-image). Reversing the order gives the opposite vector.

Angle Brackets vs. Parentheses

Vectors use angle brackets ; points use parentheses . Mixing them up confuses the reader about what you mean.

Real-Life Applications

GPS Navigation

When your GPS tells you to “head north-east for 3 miles,” it's essentially giving you a translation vector. The direction and magnitude describe exactly how your position should change on the map.

Physics — Displacement Vectors

In physics, displacement is a translation vector. An airplane flying “200 km due east then 150 km due north” is applying a vector km to its starting position.

Practice Quiz

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