Prove That Two Successive Translations Are Additive

Proving that Two Successive Translations are Additive: Understanding the Mathematical Concept

In mathematics, particularly in geometry and linear algebra, the concept of translation plays a fundamental role in understanding spatial transformations. When dealing with translations in a vector space or on a plane, it is essential to establish certain properties, such as additivity, to comprehend how these transformations interact and combine. This article delves into the mathematical proof that two successive translations are additive, exploring the underlying principles and demonstrating their application through examples.

Understanding Translations in Mathematics

Definition: A translation in mathematics refers to a transformation that moves every point of a geometric figure or vector by a fixed distance in a specified direction. Translations are characterized by their direction (vector) and magnitude (distance).

Additivity Property: The additivity property of translations states that the composition of two successive translations results in a single translation whose vector is the sum of the individual translation vectors.

Proof of Additivity for Two Successive Translations

Let’s consider two successive translations in a vector space or on a plane:

1. First Translation: Let $T_1$ be a translation that moves every point $P$ in the space by a vector $vec{v}_1$. $T_1(P) = P + vec{v}_1$

2. Second Translation: Let $T_2$ be another translation applied after $T_1$, moving every point $P$ further by a vector $vec{v}_2$. $T_2(T_1(P)) = T_2(P + vec{v}_1) = (P + vec{v}_1) + vec{v}_2$

Demonstrating Additivity

To prove that $T_2(T_1(P))$ is equivalent to a single translation $T_{text{total}}$:

$T_2(T_1(P)) = P + vec{v}_1 + vec{v}_2$

Here, $vec{v}_1 + vec{v}_2$