Additive Inverse

Definition

The additive inverse of a vector is the vector that cancels it to the zero vector.

Example:

v = [3, 5]

The additive inverse is:

-v = [-3, -5]

Because:

[3, 5] + [-3, -5] = [0, 0]

Additive inverse = the vector that brings another vector back to zero.

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Component By Component

The additive inverse flips the sign of each component.

Example:

[4, -2, 7]

Additive inverse:

[-4, 2, -7]

Each component becomes its opposite.

To find the additive inverse, reverse every component's sign.

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Relation To Subtraction

Subtraction is addition of an additive inverse.

Example:

[8, 9] - [3, 4]

is the same as:

[8, 9] + [-3, -4]

Result:

[5, 5]

Additive inverse explains vector subtraction.

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Opposite Direction

The additive inverse has the same size but opposite direction.

Example:

v  = [3, 4]
-v = [-3, -4]

Both have magnitude 5.

But they point in opposite directions.

Additive inverse preserves size while reversing direction.

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Canceling Change

If a vector represents a change, its additive inverse undoes that change.

Example:

move right and up: [3, 2]
move back:         [-3, -2]

Together:

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

The net change is zero.

Additive inverse represents the change needed to return to the starting point.

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Zero Vector Is Its Own Inverse

The zero vector's additive inverse is itself.

Example:

[0, 0] + [0, 0] = [0, 0]

There is nothing to cancel.

The zero vector already equals its own opposite.

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Final Summary

• The additive inverse cancels a vector to zero.
• It is created by flipping every component's sign.
• Subtraction is addition with an additive inverse.
• The inverse has the same magnitude but opposite direction.
• It can represent undoing a change.
• The zero vector is its own additive inverse.