Adding Vectors by Components

Adding Vectors by Components

Add horizontal parts together and vertical parts together.

Adding Vectors by Components

When several vectors act together, they can be added efficiently using their rectangular components.

Suppose vectors A⃗ and B⃗ have components:

`Aₓ, Aᵧ`

and

`Bₓ, Bᵧ`

The resultant R⃗ has:

**Rₓ = Aₓ + Bₓ**

**Rᵧ = Aᵧ + Bᵧ**

Then:

**R⃗ = Rₓî + Rᵧĵ**

Procedure

1. Choose coordinate axes. 2. Resolve each vector. 3. Assign correct signs. 4. Add all x-components. 5. Add all y-components. 6. Calculate resultant magnitude and direction if required.

Example

**A⃗ = 4î + 3ĵ**

**B⃗ = 2î − ĵ**

Then:

**R⃗ = A⃗ + B⃗**

`= (4+2)î + (3−1)ĵ`

**R⃗ = 6î + 2ĵ**

Why This Method Is Powerful

It works even when many vectors point in different directions.

Adding Vectors by Components — Advanced

Vector addition is component-wise because vectors form a linear space relative to a chosen basis.