Vector Addition (Geometric Approach): Explanation & Examples

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  • 0:02 Why We Need to Add Vectors
  • 1:21 The Geometric Approach
  • 2:41 Example
  • 3:36 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video, you will be able to explain why we might need to add two vectors and, given magnitudes and directions, add two vectors using geometric methods. A short quiz will follow.

Why We Need to Add Vectors

One day you're at a fair and are taking part in a multi-team tug-of-war. In the center is a ring with five ropes connected to it, and at the end of each rope is a tug-of-war team. Once the war begins, all five teams pull as hard as they can. Some pull with large forces, some with smaller forces. But which direction does the ring move? Which way does it accelerate?

To figure out the answer to that question, we need to find the sum total of the force vectors of each of the teams. If two teams pull in opposite directions with the same force, they'll cancel out, but with the number of forces that can be involved in real-life situations, there's no telling what the overall force will be. To figure out the overall force, we have to add up the force vectors.

A similar thing can happen with other kinds of vectors. If you're walking on a moving train, while the earth itself is moving around the sun and the sun is orbiting around the center of the galaxy, your body has a lot of different velocity vectors. If you wanted to know your total velocity, you would have to add those up.

There are two main ways of adding vectors: mathematically and geometrically. In this lesson, we're going to go through the geometric approach: adding vectors tip to tail.

The Geometric Approach

To avoid lots of messy math, the geometric approach has a nice simplicity to it. All you have to do is draw a diagram. Of course, you do have to draw a very accurate diagram - a scale drawing. A scale drawing is a drawing where the lengths and angles of every line relate to each other in a consistent way that matches the reality. So for example, with a force vector, you could say that every centimeter equals a force of 10 newtons. So a 50 newton force vector would be an arrow of 5 centimeters in length.

First diagram
vector diagram for example

To get the total of two vectors, you draw them to scale, and put the tail (backside of the arrow) of one vector against the tip (the point) of the other vector. If, for example, we have two force vectors pushing on a block, one person is pushing the block down and to the right, the other down and to the left. We move one of these vectors onto the end of the other.

Second diagram
vector diagram for example

And if this diagram is to scale, we can find the total by drawing a vector and arrow from the very start to the very finish. This vector we've drawn is the total. The vector for force 1 plus the vector for force 2 equals force-total. We could then measure the length of the arrow in our scale drawing and the angle in which the arrow is pointing to figure out some numbers for our total force.

Final diagram
vector diagram for example

Example

Maybe this would be easier if we went through an example. Let's go back to the 5-way tug of war, but let's make it 3 teams to be simpler. Here is a scale drawing of the forces on the ring in the center:

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