Subtraction and Addition of Vectors: Methods, Formulas, Videos ... - Free Printable

The image illustrates the addition and subtraction of vectors using geometric methods (specifically, the triangle method). Let's go through each part step by step and explain how vector addition and subtraction work.

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🔷 1. Vector Addition: $ \vec{a} + \vec{b} $

Diagram on the left:
- Two vectors $ \vec{a} $ and $ \vec{b} $ are shown.
- $ \vec{a} $ is drawn first (blue), then $ \vec{b} $ is placed head-to-tail with $ \vec{a} $ (red).
- The resultant vector $ \vec{a} + \vec{b} $ is the vector that starts from the tail of $ \vec{a} $ and ends at the head of $ \vec{b} $.
- This follows the triangle law of vector addition.

✔ Rule:
To add two vectors, place them head-to-tail; the sum is the vector from the starting point to the final endpoint.

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🔷 2. Vector Subtraction: $ \vec{b} - \vec{a} $

Middle diagram:
- We want $ \vec{b} - \vec{a} $, which is equivalent to $ \vec{b} + (-\vec{a}) $.
- $ -\vec{a} $ is the vector $ \vec{a} $ but in the opposite direction.
- So, we draw $ \vec{a} $, then reverse it to get $ -\vec{a} $, and then add $ \vec{b} $ to it.
- But here, the diagram shows:
- Start with $ \vec{a} $ (blue), then draw $ \vec{b} $ (red) from the tip of $ \vec{a} $.
- The vector $ \vec{b} - \vec{a} $ is drawn from the tip of $ \vec{a} $ to the tip of $ \vec{b} $ — this represents the difference $ \vec{b} - \vec{a} $.

✔ Alternative Rule:
$ \vec{b} - \vec{a} $ is the vector that goes from the tip of $ \vec{a} $ to the tip of $ \vec{b} $ when both vectors start from the same point.

> Alternatively, you can think of it as:
> $ \vec{b} - \vec{a} = \vec{b} + (-\vec{a}) $

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🔷 3. Vector Subtraction: $ \vec{a} - \vec{b} $

Right diagram:
- Similar logic applies: $ \vec{a} - \vec{b} = \vec{a} + (-\vec{b}) $
- Here, $ \vec{b} $ is reversed (i.e., $ -\vec{b} $), and added to $ \vec{a} $.
- The result $ \vec{a} - \vec{b} $ is drawn from the tip of $ \vec{b} $ to the tip of $ \vec{a} $ when both originate from the same point.

✔ Note:
$ \vec{a} - \vec{b} $ and $ \vec{b} - \vec{a} $ are equal in magnitude but opposite in direction:
$$
\vec{a} - \vec{b} = -(\vec{b} - \vec{a})
$$

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✔ Summary:

| Operation | Method |
|------------------|------------------------------------------------------------------------|
| $ \vec{a} + \vec{b} $ | Place $ \vec{b} $ head-to-tail with $ \vec{a} $; resultant is from tail of $ \vec{a} $ to head of $ \vec{b} $. |
| $ \vec{b} - \vec{a} $ | Draw $ \vec{a} $ and $ \vec{b} $ from same origin; $ \vec{b} - \vec{a} $ is from tip of $ \vec{a} $ to tip of $ \vec{b} $. |
| $ \vec{a} - \vec{b} $ | Same as above, but from tip of $ \vec{b} $ to tip of $ \vec{a} $. |

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📌 Key Points:

- Vector addition is commutative: $ \vec{a} + \vec{b} = \vec{b} + \vec{a} $
- Vector subtraction is not commutative: $ \vec{a} - \vec{b} \neq \vec{b} - \vec{a} $
- The magnitude of $ \vec{a} - \vec{b} $ is the same as $ \vec{b} - \vec{a} $, but the direction is opposite.

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Let me know if you'd like a numerical example or a diagram explanation!