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Vector Addition/Subtraction (Component Form) Calculator

Calculates the resultant vector's components by adding or subtracting the corresponding components of individual vectors.

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Result
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R (x-component)

Formula first

Overview

This equation provides a straightforward method for combining vectors in 2D or 3D space by operating on their individual Cartesian components. For vector addition, the x, y, and z components of the resultant vector are found by summing the respective components of the input vectors. For subtraction, the components are subtracted. This component-wise approach simplifies vector operations, making them analogous to scalar arithmetic and is fundamental in physics and engineering.

Symbols

Variables

A_x = A (x-component), A_y = A (y-component), A_z = A (z-component), B_x = B (x-component), B_y = B (y-component)

A (x-component)
A (y-component)
A (z-component)
B (x-component)
B (y-component)
B (z-component)
R (x-component)
R (y-component)
R (z-component)

Apply it well

When To Use

When to use: Use this formula when you need to find the resultant vector from two or more vectors, and those vectors are given in component form (e.g., $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$). It's particularly useful for problems involving forces, velocities, or displacements in multiple dimensions.

Why it matters: Vector addition and subtraction are foundational operations in physics and engineering, enabling the analysis of complex systems. From calculating the net force on an object to determining the trajectory of a projectile or the resultant velocity of an aircraft in wind, understanding how to combine vectors in component form is essential for solving real-world problems.

Avoid these traps

Common Mistakes

  • Mixing up components (e.g., adding to ).
  • Incorrectly handling negative signs during subtraction.
  • Forgetting that vector subtraction is not commutative ().
  • Not specifying the operation (addition or subtraction) clearly.

One free problem

Practice Problem

Given vector and vector , calculate the x-component of the resultant vector .

A (x-component)3 m
A (y-component)-2 m
A (z-component)5 m
B (x-component)1 m
B (y-component)4 m
B (z-component)-3 m

Solve for:

Hint: Add the corresponding x-components of vectors A and B.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Halliday, Resnick, Walker, Fundamentals of Physics, 11th ed.
  2. Bird, Stewart, Lightfoot, Transport Phenomena, 2nd ed.
  3. Wikipedia: Vector addition
  4. Halliday, Resnick, Walker, Fundamentals of Physics
  5. Bird, Stewart, Lightfoot, Transport Phenomena
  6. Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics. 11th ed. Wiley, 2018.
  7. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. Transport Phenomena. 2nd ed. Wiley, 2007.
  8. Wikipedia: Euclidean vector