Vector Addition

DYUP

What is it?

Vector addition combines two or more vectors to find a single vector that shows their total effect. Think of it like combining directions on a map—each vector has a size and a direction. You can add them by aligning them tip-to-tail or breaking them into parts (like north/south and east/west). The final result, called the resultant vector, shows the overall direction and strength of all the vectors combined.

How To Solve

1. Understand the Problem: Identify the vectors, including their magnitudes (size) and directions (angles).

2. Break Down Each Vector: Use trigonometry to calculate the x and y components for each vector:

x = magnitude × cos(angle)
y = magnitude × sin(angle)

3. Add Components: Add all x-components to find the total Σx , and do the same for the y-components to find Σy.

4. Find the Resultant Vector: Calculate the magnitude using the Pythagorean theorem:

Resultant Magnitude = √(Σx² + Σy²)

Find the angle using:

Angle = tan⁻¹(Σy / Σx)

5. Express the Result: Write the resultant vector as magnitude and direction, e.g., “The resultant is 10 m/s at 45°.”

Eqautions

1. Component Formulas:

x = magnitude × cos(angle)
y = magnitude × sin(angle)

2. Resultant Magnitude:

Resultant Magnitude = √(Σx² + Σy²)

3. Resultant Angle:

Resultant Angle = tan⁻¹(Σy / Σx)

4. Adding Multiple Vectors:

Σx = x₁ + x₂ + ... + xₙ
Σy = y₁ + y₂ + ... + yₙ

5. Resultant Vector in Polar Form:

Vector Magnitude = √(x² + y²)
Vector Direction = tan⁻¹(y / x)

6. Unit Vector Representation:

v = x î + y ĵ

Vector Addition Calculator

Magnitude: Direction (°):

Result:

Entered Vectors: