Vector Addition/Subtraction (Component Form) Equation, Derivation & Rearrangements

Core idea

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.

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.

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}

A (x-component)

m

A_{y}

A (y-component)

m

A_{z}

A (z-component)

m

B_{x}

B (x-component)

m

B_{y}

B (y-component)

m

B_{z}

B (z-component)

m

R_{x}

R (x-component)

m

R_{y}

R (y-component)

m

R_{z}

R (z-component)

m

Walkthrough

Derivation

Formula: Vector Addition/Subtraction (Component Form)

Vector addition and subtraction in component form are performed by adding or subtracting the corresponding scalar components.

Vectors are expressed in a common Cartesian coordinate system.
The components are scalar values representing the projection of the vector onto each axis.

1

Define Vectors in Component Form:

Represent each vector as the sum of its components along the x, y, and z axes, where $i$ , $j$ , $k$ are unit vectors in those directions.

A = A_{x} i + A_{y} j + A_{z} k B = B_{x} i + B_{y} j + B_{z} k

2

Vector Addition:

To add two vectors, simply add their corresponding components. The resultant vector $R$ will have its x-component as the sum of the x-components, its y-component as the sum of the y-components, and so on.

R = A + B = (A_{x} + B_{x}) i + (A_{y} + B_{y}) j + (A_{z} + B_{z}) k

3

Vector Subtraction:

To subtract vector $B$ from vector $A$ , subtract the corresponding components of $B$ from those of $A$ . This is equivalent to adding $A$ to the negative of $B$ (where $- B$ has components $- B_{x}, - B_{y}, - B_{z}$ ).

R = A - B = (A_{x} - B_{x}) i + (A_{y} - B_{y}) j + (A_{z} - B_{z}) k

Note: Pay careful attention to the order of subtraction and the signs of the components.

Result

R = A - B = (A_{x} - B_{x}) i + (A_{y} - B_{y}) j + (A_{z} - B_{z}) k

Source: Cambridge International AS & A Level Mathematics: Pure Mathematics 1 (Chapter 11: Vectors)

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a linear line with a slope of 1, represented by the equation R_x = x + B_x, where the y-intercept is determined by the constant value of B_x. For a student, this linear relationship means that increasing the x-component of vector A results in an identical increase in the resultant x-component, regardless of whether the starting value is large or small. The most important feature is that the constant slope of 1 shows a direct one-to-one correspondence between the input and the output.

Graph type: linear

Why it behaves this way

Intuition

Visually, vector addition means placing vectors head-to-tail and drawing the resultant from the first tail to the last head, while subtraction is equivalent to adding the negative (reversed) second vector.

R

The resultant vector, representing the combined effect (sum or difference) of the input vectors.

This is the single vector that summarizes the net outcome when multiple vector influences are considered.

A_{x}

The scalar component of vector \mathbf{A} along the x-axis. Similar interpretations apply to A_y, A_z, B_x, B_y, and B_z for their respective vectors and axes.

This number tells you how much of the vector's magnitude and direction is aligned purely along one of the primary coordinate axes.

i

A unit vector pointing in the positive x-direction. Similar interpretations apply to \mathbf{j} for the y-direction and \mathbf{k} for the z-direction.

These act as directional labels, indicating which component corresponds to which axis in the coordinate system.

Signs and relationships

±: The plus sign (+) indicates vector addition, where corresponding components are summed. The minus sign (-) indicates vector subtraction, where the components of the second vector are subtracted from the first, which is

Free study cues

Insight

Canonical usage

The components of the resultant vector will have the same units as the components of the input vectors, as all vectors involved must represent the same physical quantity.

Common confusion

A common mistake is attempting to add or subtract vectors that represent different physical quantities (e.g., adding a displacement vector to a force vector), which is dimensionally incorrect.

Unit systems

$A$ Any consistent unit (e.g., m, N, m/s) · Vector A must represent a specific physical quantity (e.g., displacement, force, velocity). All its components (A_x, A_y, A_z) must share the same unit and dimension.

$B$ Same unit as vector A · Vector B must represent the same physical quantity as vector A. All its components (B_x, B_y, B_z) must share the same unit and dimension, which must be consistent with vector A's components.

$R$ Same unit as vector A and B · The resultant vector R will represent the same physical quantity as vectors A and B, and its components (R_x, R_y, R_z) will have the same units and dimensions as the components of A and B.

One free problem

Practice Problem

Given vector $A = 3 i - 2 j + 5 k$ and vector $B = 1 i + 4 j - 3 k$ , calculate the x-component of the resultant vector $R = A + B$ .

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: $R x$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the resultant force acting on an airplane by summing the lift, drag, thrust, and weight vectors, each expressed in their x, y, z components.

Study smarter

Tips

Ensure you add/subtract corresponding components (x with x, y with y, z with z).
Pay close attention to the signs of the components, especially during subtraction.
Remember that the resultant is also a vector, expressed in component form.
If vectors are not initially in component form, resolve them into components first.

Avoid these traps

Common Mistakes

Mixing up components (e.g., adding $A_{x}$ to $B_{y}$ ).
Incorrectly handling negative signs during subtraction.
Forgetting that vector subtraction is not commutative ( $A - B \neq = B - A$ ).
Not specifying the operation (addition or subtraction) clearly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Vector addition and subtraction in component form are performed by adding or subtracting the corresponding scalar components.

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.

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.

Mixing up components (e.g., adding A_x to B_y). Incorrectly handling negative signs during subtraction. Forgetting that vector subtraction is not commutative ($\mathbf{A} - \mathbf{B} \neq \mathbf{B} - \mathbf{A}$). Not specifying the operation (addition or subtraction) clearly.