Vector Subtraction

Definition

Vector subtraction subtracts matching components.

Example:

[7, 9] - [2, 4] = [5, 5]

Because:

7 - 2 = 5
9 - 4 = 5

Vector subtraction = component-by-component difference.

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Same Dimension Requirement

Like addition, subtraction requires the same dimension.

Valid:

[7, 9] - [2, 4]

Invalid:

[7, 9] - [2, 4, 1]

The components do not line up.

Subtraction requires matching structure.

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Subtraction As Difference

Subtraction can show how one vector differs from another.

Example:

current = [8, 10]
target  = [5, 4]

Difference:

current - target = [3, 6]

This means:

first component differs by 3
second component differs by 6

Vector subtraction exposes component-level differences.

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Subtraction As Movement

If two vectors represent positions, subtraction can describe movement between them.

Example:

start = [2, 3]
end   = [7, 8]

Movement from start to end:

end - start = [5, 5]

So [5, 5] is the change needed to get from the first position to the second.

Subtraction can produce a change vector.

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Order Matters

Vector subtraction is not the same in both directions.

Example:

[7, 9] - [2, 4] = [5, 5]
[2, 4] - [7, 9] = [-5, -5]

The size of the difference may match, but the direction is opposite.

Subtraction keeps track of direction of change.

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Relation To Addition

Subtraction can be understood as adding the opposite vector.

Example:

[7, 9] - [2, 4]

is the same as:

[7, 9] + [-2, -4]

Result:

[5, 5]

Subtraction is addition with an additive inverse.

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Final Summary

• Vector subtraction subtracts matching components.
• Vectors must have the same dimension.
• Subtraction can measure difference.
• Subtraction can describe movement from one vector to another.
• Order matters.
• Subtraction is connected to additive inverse.