Vector addition combines two or more vectors to find a single resultant vector, often using the graphical head-to-tail method. Vector subtraction is a special case of addition where you add the negative of the second vector.

Adding Vectors: Head-to-Tail Method

The easiest way to add vectors graphically is the head-to-tail method.

  1. Draw the first vector: Start at a point and draw the first vector (let’s call it Vector A) with its correct length and direction.
  2. Draw the second vector: From the head (arrow part) of Vector A, draw the tail of the second vector (Vector B) with its correct length and direction.
  3. Draw the Resultant: The resultant vector (which is A + B) is drawn from the tail of the first vector (A) to the head of the last vector (B).

The resultant vector shows the combined effect of the two original vectors.

Interactive: Vector Addition

Let’s see how to add two vectors, Vector A and Vector B!

Interactive Vector Addition and Subtraction An animation demonstrating how to add and subtract vectors using graphical methods. Vector A Vector B Vector B A + B -B -B A - B

Click "Next Step" to learn how to add and subtract vectors!

Subtracting Vectors: Adding the Negative

Subtracting vectors might sound tricky, but it’s actually just a special kind of addition!

To subtract Vector B from Vector A (A - B), you simply add the negative of Vector B to Vector A.

  • What is the negative of a vector? The negative of a vector (e.g., -B) has the same magnitude (length) as the original vector (B), but points in the exact opposite direction.
  • Once you find -B, you use the same head-to-tail method to add A + (-B).

Why Vector Addition and Subtraction Matter

  • Combining Forces: If multiple forces push or pull on an object, vector addition helps you find the single “net force” that tells you how the object will move.
  • Navigation: If you travel in different directions, vector addition helps you find your final displacement (how far and in what direction you are from your starting point).
  • Relative Motion: Understanding how velocities combine (like a boat moving in a river) uses vector addition.

Interactive Match: Vector Addition & Subtraction

Test your understanding of key terms and methods related to combining vectors.

Click a term and then its matching meaning. Match all pairs to complete!

Audio Explanation

Prefer to listen? Here's a quick audio summary of vector addition and subtraction.

💡 Quick Concept Check:

If you walk 3 meters East, then 4 meters North, what is your total displacement (magnitude and direction)?

Click to Reveal Answer
Your total displacement is 5 meters, 53.1 degrees North of East. This is found by adding the two displacement vectors using the head-to-tail method (forming a right triangle).

Ready to put your understanding of vector addition and subtraction into practice? Check out these related skills:

Practice Problems

Test your understanding and apply what you've learned with these problems.

↑ Back to top