Vector Addition

Combine two vectors by adding their components separately to find the resultant vector that represents their total effect or displacement.

u+v=⟨u1+v1,;u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, ; u_2 + v_2 \rangle

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What each symbol means

What Vector Addition takes
u\mathbf{u}
v\mathbf{v}
Vector Addition →
SymbolMeaning
u\mathbf{u}The first vector being added, written in bold to denote it has both magnitude and direction; treating it as a scalar (a single number) ignores the directional component and gives a wrong resultant.
v\mathbf{v}The second vector being added, in the same dimension as u\mathbf{u}; if confused with a scalar or if its dimension differs from u\mathbf{u}, the addition is undefined or incorrect.

When to use it

Use vector addition when you need to find the total effect of two or more vectors acting together.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Add two vectors with positive integers

Problem

Add the vectors u=⟨2,3⟩\mathbf{u} = \langle 2, 3 \rangle and v=⟨4,1⟩\mathbf{v} = \langle 4, 1 \rangle.
  1. u1+v1=2+4=6u_1 + v_1 = 2 + 4 = 6

    Add the first components; vector addition works component-by-component.

  2. u2+v2=3+1=4u_2 + v_2 = 3 + 1 = 4

    Add the second components using the same rule.

  3. u+v=⟨6,4⟩\mathbf{u} + \mathbf{v} = \langle 6, 4 \rangle

    The result is a vector with the two sums as components.

Answer: ⟨6,4⟩\langle 6, 4 \rangle

Vector addition combines the components separately. The first component of the sum is the sum of the first components, and the second component is the sum of the second components. This gives us the resultant vector that represents both forces or displacements acting together.

2. Add vectors with negative components

Problem

Add the vectors u=⟨−3,5⟩\mathbf{u} = \langle -3, 5 \rangle and v=⟨2,−2⟩\mathbf{v} = \langle 2, -2 \rangle.
  1. u1+v1=−3+2=−1u_1 + v_1 = -3 + 2 = -1

    Add the first components, treating negatives as signed numbers: −3+2=−1-3 + 2 = -1.

  2. u2+v2=5+(−2)=3u_2 + v_2 = 5 + (-2) = 3

    Add the second components; the negative component represents direction opposite to the positive one.

  3. u+v=⟨−1,3⟩\mathbf{u} + \mathbf{v} = \langle -1, 3 \rangle

    The resultant vector has a negative first component and positive second component.

Answer: ⟨−1,3⟩\langle -1, 3 \rangle

When vectors contain negative components, treat them as signed numbers. Add them using standard integer addition rules. The negative components often represent opposite directions, so the resultant can point in a different quadrant than the original vectors.

3. Find total force on a bake sale tray

Problem

At a school bake sale, Sarah carries a tray with a force of ⟨10,5⟩\langle 10, 5 \rangle pounds (forward and upward), and adds more items creating an additional force of ⟨3,2⟩\langle 3, 2 \rangle pounds in the same directions. What is the total force on the tray?
  1. F1total=10+3=13F_1^{\text{total}} = 10 + 3 = 13

    Add the forward-direction forces because both act in the same direction.

  2. F2total=5+2=7F_2^{\text{total}} = 5 + 2 = 7

    Add the upward-direction forces using the same principle.

  3. Ftotal=⟨13,7⟩ pounds\mathbf{F}_{\text{total}} = \langle 13, 7 \rangle \text{ pounds}

    The total force is a vector combining both the forward and upward effects.

Answer: ⟨13,7⟩ pounds\langle 13, 7 \rangle \text{ pounds}

In real-world situations, multiple forces act on objects. Vector addition lets us find the net effect by combining the components in each direction. Here, adding the forward and upward components separately gives us the total force pushing on the tray.

Common mistakes

Where Vector Addition usually goes wrong
Answer came out wrong
Adding all four numbers to get u+v=⟨2+3+4+1⟩=⟨10⟩\mathbf{u} + \mathbf{v} = \langle 2 + 3 + 4 + 1 \rangle = \langle 10 \rangle
Add the corresponding components: u1+v1=2+4=6u_1 + v_1 = 2 + 4 = 6 and u2+v2=3+1=4u_2 + v_2 = 3 + 1 = 4 to get ⟨6,4⟩\langle 6, 4 \rangle.
Multiplying the components: u+v=⟨2⋅4,3⋅1⟩=⟨8,3⟩\mathbf{u} + \mathbf{v} = \langle 2 \cdot 4, 3 \cdot 1 \rangle = \langle 8, 3 \rangle
Use addition only: u+v=⟨2+4,3+1⟩=⟨6,4⟩\mathbf{u} + \mathbf{v} = \langle 2 + 4, 3 + 1 \rangle = \langle 6, 4 \rangle.
Adding only the first components and ignoring the second: u+v=⟨6⟩\mathbf{u} + \mathbf{v} = \langle 6 \rangle
Add each component: first 2+4=62 + 4 = 6, then 3+1=43 + 1 = 4, giving ⟨6,4⟩\langle 6, 4 \rangle.
The mistakeWhy it is wrongThe fix
Adding all four numbers to get u+v=⟨2+3+4+1⟩=⟨10⟩\mathbf{u} + \mathbf{v} = \langle 2 + 3 + 4 + 1 \rangle = \langle 10 \rangleVectors are added component-by-component, not by summing all components into one.Add the corresponding components: u1+v1=2+4=6u_1 + v_1 = 2 + 4 = 6 and u2+v2=3+1=4u_2 + v_2 = 3 + 1 = 4 to get ⟨6,4⟩\langle 6, 4 \rangle.
Multiplying the components: u+v=⟨2⋅4,3⋅1⟩=⟨8,3⟩\mathbf{u} + \mathbf{v} = \langle 2 \cdot 4, 3 \cdot 1 \rangle = \langle 8, 3 \rangleThe operation is addition, not multiplication; confusing it with the dot product (which yields a scalar) or scalar multiplication leads to wrong answers.Use addition only: u+v=⟨2+4,3+1⟩=⟨6,4⟩\mathbf{u} + \mathbf{v} = \langle 2 + 4, 3 + 1 \rangle = \langle 6, 4 \rangle.
Adding only the first components and ignoring the second: u+v=⟨6⟩\mathbf{u} + \mathbf{v} = \langle 6 \rangleBoth components must be added separately to preserve the vector's dimensionality and directional information.Add each component: first 2+4=62 + 4 = 6, then 3+1=43 + 1 = 4, giving ⟨6,4⟩\langle 6, 4 \rangle.

Tips and when to use something else

  • Always add component-by-component: line up the first components and add them, then line up the second components and add them.
  • Vector addition only works if both vectors have the same dimension; you cannot add a 2D vector to a 3D vector.
  • If you need a single number (like distance or work done), use the dot product instead of vector addition; vector addition always returns another vector.
  • Check your answer by counting: the sum of two 2D vectors should be a 2D vector, not a single number.

Related formulas

Magnitude of a VectorMagnitude of a vector is its length or size, measured as the distance from the origin; use it to find how far something has traveled overall.Dot ProductThe dot product is a scalar value that measures how much two vectors align; use it to find angles between vectors or project one vector onto another.Vector ProjectionVector projection breaks down one vector into a component along another's direction, essential for decomposing forces in physics and engineering.

Frequently asked questions

Why add components separately and not all together?
Vectors represent directions as well as magnitudes. Adding components separately preserves both pieces of information—which is why we get a vector back, not a single number. Adding all four numbers together destroys the directional structure.
Can you add vectors of different dimensions, like a 2D vector and a 3D vector?
No. Both vectors must have the same number of components. A 2D vector has two components (left-right and up-down), while a 3D vector has three (left-right, up-down, and forward-back). Mixing them is like trying to add an apple and an orange—the dimensions have to match.
What's the difference between vector addition and the dot product?
Vector addition combines two vectors to produce a third vector (with the same dimension). The dot product multiplies them together to produce a single number (a scalar). Use vector addition when you need the resultant vector; use the dot product when you need a measure of how aligned the vectors are.
Does the order matter? Is u+v\mathbf{u} + \mathbf{v} the same as v+u\mathbf{v} + \mathbf{u}?
Yes, order does not matter. Vector addition is commutative, meaning u+v=v+u\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}. This makes sense physically: pushing forward then right gives the same result as pushing right then forward.

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Reviewed 2026-09-18