Three displacements are 𝐀=200 m due south, 𝐁=250 m due...

Three displacements are $\overrightarrow{\mathbf{A}}=200 \mathrm{~m}$ due south, $\overrightarrow{\mathbf{B}}=250 \mathrm{~m}$ due west, and $\overrightarrow{\mathbf{C}}=150 \mathrm{~m}$ at $30.0^{\circ}$ east of north. (a) Construct a separate diagram for each of the following possible ways of adding these vectors: $\overrightarrow{\mathbf{R}}_{1}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}} ; \overrightarrow{\mathbf{R}}_{2}=\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{A}}$ $\overrightarrow{\mathbf{R}}_{3}=\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{A}}$. (b) Explain what you can conclude from comparing the diagrams.

Three displacements are $\overrightarrow{\mathbf{A}}=200 \mathrm{~m}$ due south, $\overrightarrow{\mathbf{B}}=250 \mathrm{~m}$ due west, and $\overrightarrow{\mathbf{C}}=150 \mathrm{~m}$ at $30.0^{\circ}$ east of north. (a) Construct a separate diagram for each of the following possible ways of adding these vectors: $\overrightarrow{\mathbf{R}}_{1}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}} ; \overrightarrow{\mathbf{R}}_{2}=\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{A}}$
$\overrightarrow{\mathbf{R}}_{3}=\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{A}}$. (b) Explain what you can conclude from comparing the diagrams.

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Key Concepts

Graphical Representation of Vectors

Graphical methods such as using vector diagrams or the head-to-tail method are essential for visualizing the process of vector addition. Diagrams help in understanding how different vectors combine spatially, allowing for the illustration of properties like the commutative nature of vector addition.

Displacement Vectors

Displacement vectors represent both the distance and the specific direction from an initial to a final position. Understanding displacement is crucial in connecting vector addition to real-world movement, as it encapsulates the concept of net change in position through the combination of individual movements.

Vector Addition

This concept involves combining two or more vectors to produce a single resultant vector. It is foundational in physics and engineering for determining the overall effect of several displacements or forces, taking into account both magnitude and direction.

Commutative Property of Vector Addition

The commutative property states that when adding vectors, the order in which they are added does not change the resultant vector. This means that regardless of whether a vector is added first or last, the final summed vector remains the same in both magnitude and direction.

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