Vector Addition

Definition

Vector addition combines two vectors by adding matching components.

Example:

[2, 3] + [4, 5] = [6, 8]

Because:

2 + 4 = 6
3 + 5 = 8

Vector addition = component-by-component addition.

---

Matching Positions

Only matching positions are added.

Example:

[a, b] + [c, d] = [a + c, b + d]

The first component combines with the first component.

The second component combines with the second component.

Position controls what gets added together.

---

Same Dimension Requirement

Vectors must have the same dimension to be added.

Valid:

[1, 2] + [3, 4] = [4, 6]

Invalid:

[1, 2] + [3, 4, 5]

There is no matching component for every position.

Vector addition requires aligned structure.

---

Addition As Combining Information

If vectors represent measurements, addition combines the measurements.

Example:

morning sales = [2, 5]
evening sales = [3, 4]

Total:

[2, 5] + [3, 4] = [5, 9]

The first category total is 5.

The second category total is 9.

Vector addition combines corresponding parts of structured data.

---

Addition As Movement

A vector can also represent movement.

Example:

start = [0, 0]
move  = [3, 2]

New position:

[0, 0] + [3, 2] = [3, 2]

Another move:

[3, 2] + [1, 4] = [4, 6]

Adding a vector can update a position or state.

---

Order Does Not Change The Result

For vector addition:

[2, 3] + [4, 5] = [6, 8]
[4, 5] + [2, 3] = [6, 8]

Changing the order does not change the result.

This works because scalar addition has the same property.

Vector addition inherits structure from scalar addition.

---

Final Summary

• Vector addition adds matching components.
• Vectors must have the same dimension.
• Component order must be consistent.
• Addition can combine data or update a state.
• Vector addition preserves vector structure.
• The result has the same dimension as the input vectors.