Which Vector Is The Sum Of The Vectors Shown Below

Introduction

When you are asked which vector is the sum of the vectors shown below, the task is essentially to determine the resultant vector that results from adding two or more independent vectors together. Still, this question appears frequently in physics, engineering, and mathematics courses because vector addition is a fundamental operation that describes how forces, velocities, or displacements combine in the real world. In this article we will walk through a clear, step‑by‑step process for finding the correct vector, explain the underlying scientific principles, and answer common questions that students often have. By the end, you will be able to look at any set of vectors, perform the addition accurately, and confidently identify the vector that represents the sum Worth knowing..

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Steps

To answer which vector is the sum of the vectors shown below, follow these organized steps. Each step is presented as a sub‑heading (H3) for easy reference Most people skip this — try not to..

1. Identify the components of each vector

• Component method – Break every vector into its horizontal (x) and vertical (y) components.
• If the vectors are given graphically, use a ruler and protractor to read the magnitude and direction, then convert to components with:
  • x‑component = magnitude × cos(angle)
  • y‑component = magnitude × sin(angle)

Why this matters: Adding vectors directly is difficult when they are drawn at various angles. Working with components lets you use simple arithmetic Most people skip this — try not to..

2. Add the horizontal components together

• Sum all x‑components to obtain the total horizontal displacement (X_total).
• Write the result as a single number; keep the sign (positive or negative) to indicate direction.

3. Add the vertical components together

• Sum all y‑components to obtain the total vertical displacement (Y_total).
• As with the x‑components, retain the sign for direction.

4. Form the resultant vector

• The resultant vector (R) has components (X_total, Y_total).
• You can represent R in three common ways:
  1. Component form – R = (X_total, Y_total)
  2. Magnitude‑direction form – calculate magnitude = √(X_total² + Y_total²) and direction = atan2(Y_total, X_total)
  3. Graphical form – draw the vectors head‑to‑tail; the vector that starts at the tail of the first and ends at the head of the last is the sum.

5. Match the resultant to the labeled vectors

• Compare the calculated components (or magnitude‑direction) with the vectors shown in the figure.
• The vector whose components (or length and angle) match R is the vector that is the sum of the vectors shown below.

Tip: If the figure provides the vectors as arrows starting from a common origin, simply place them tip‑to‑tail in the order given; the final arrow represents the sum Turns out it matters..

Scientific Explanation

Understanding why vector addition works requires a look at the underlying geometry and algebra Easy to understand, harder to ignore..

Vector addition as a geometric operation

Vectors are represented by directed line segments. So the head‑to‑tail method is a direct consequence of the parallelogram law: if you draw a parallelogram using two vectors as adjacent sides, the diagonal of the parallelogram is the sum of those vectors. This law extends naturally to more than two vectors: you can chain them sequentially, and the overall displacement is the vector that connects the starting point to the final endpoint.

Algebraic justification

When vectors are expressed in component form, addition follows