The Addition of Two Vectors: A Visual Guide

[parshyaa]

why addition of two vectors are represented by this diagram, why the sum of two vectors are between both the vectors.

• Does it takes the idea of hitting a ball , if we hit a ball to its left side it goes right side and when hitted to its right side it goes left side and when we hit simultaneously it will go in between left and right.

 

[fresh_42]

Yes. You may imagine a vector being a force, that points in a certain direction. The strength of the force is represented by the length of the vector. E.g. a sailing ship is driven by the wind into one direction and the rudder adds a force by the resistance of water into eventually another direction, forcing the ship into a different direction than the wind blows.

Mathematically you could either use coordinates and see where addition of them leads you to, or think of a vector as simply being an arrow. If you want to add two of them, they have to be somehow related to each other. To solve this you apply them at the same point in space. This automatically spans a parallelogram. Now you simply define its diagonal as the sum of the two vectors. Some considerations about the sailing ship or the coordinate version of the vectors show, that it perfectly makes sense to do so.

 

[jack action]

Look at it like someone walking a path. All paths are judged equivalent because they all begin at Q and end at S. But it could be a path much more complicated and it would also be equivalent, like in the following image:

 

[BvU]

Hitting the ball is an analogy. Moving the ball is another (somewhat easier and quieter) analogy
(after all, the word vector comes from vehere which means 'move' ):

If we move the ball from O by vector $\vec A$ it ends up in R
If we then move the ball from O by vector $\vec B$ it ends up in S

If we move the ball from O by vector $\vec B$ it ends up in A
If we then move the ball from O by vector $\vec A$ it ends up in S

(whichs shows that $\vec A + \vec B = \vec B + \vec A$).​

We designate $\alpha$ as the angle between $\vec A$ and $\vec B$ and use a little Pythagoras. Then it's easy to see that $$
|\vec A + \vec B |^2 = \left ( |\vec A| + |\vec B |\cos\alpha\right) ^2 + \left ( |\vec B |\sin\alpha\right) ^2 = |\vec A|^2 + |\vec B |^2 + 2 |\vec A| |\vec B |\cos\alpha
$$
For more, check here