Dot Product Tools for R² and R³

The dot product can be understood through coordinates, lengths, and distances. These notes show how the dot product formula appears naturally from expanding the squared distance between two vectors.

For R²

Let

A = (x₁, y₁)

and

B = (x₂, y₂).

The length of A is

|A| = (x₁² + y₁²)¹ᐟ²,

so

|A|² = x₁² + y₁².

Similarly,

|B| = (x₂² + y₂²)¹ᐟ²,

so

|B|² = x₂² + y₂².

The dot product is

A · B = x₁x₂ + y₁y₂.

Now consider the vector from A to B:

B − A = (x₂ − x₁, y₂ − y₁).

Therefore,

|B − A| = ((x₂ − x₁)² + (y₂ − y₁)²)¹ᐟ²,

and

|B − A|² = (x₂ − x₁)² + (y₂ − y₁)².

Expanding gives

|B − A|² = x₂² − 2x₁x₂ + x₁² + y₂² − 2y₁y₂ + y₁².

Rearranging,

|B − A|² = (x₁² + y₁²) + (x₂² + y₂²) − 2(x₁x₂ + y₁y₂).

Using the length and dot product formulas,

|B − A|² = |A|² + |B|² − 2A · B.

For R³

Let

A = (x₁, y₁, z₁)

and

B = (x₂, y₂, z₂).

The length of A is

|A| = (x₁² + y₁² + z₁²)¹ᐟ²,

so

|A|² = x₁² + y₁² + z₁².

Similarly,

|B| = (x₂² + y₂² + z₂²)¹ᐟ²,

so

|B|² = x₂² + y₂² + z₂².

The dot product is

A · B = x₁x₂ + y₁y₂ + z₁z₂.

Now consider the vector from A to B:

B − A = (x₂ − x₁, y₂ − y₁, z₂ − z₁).

Therefore,

|B − A| = ((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)¹ᐟ²,

and

|B − A|² = (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)².

Expanding gives

|B − A|² = x₂² − 2x₁x₂ + x₁² + y₂² − 2y₁y₂ + y₁² + z₂² − 2z₁z₂ + z₁².

Rearranging,

|B − A|² = (x₁² + y₁² + z₁²) + (x₂² + y₂² + z₂²) − 2(x₁x₂ + y₁y₂ + z₁z₂).

Using the length and dot product formulas,

|B − A|² = |A|² + |B|² − 2A · B.

The Main Tool

From

|B − A|² = |A|² + |B|² − 2A · B,

we can rearrange to get

2A · B = |A|² + |B|² − |B − A|².

Therefore,

A · B = (|A|² + |B|² − |B − A|²) / 2.

Conclusion

The dot product can be recovered from three squared lengths:

|A|², |B|², and |B − A|².

So the dot product is not just a coordinate formula. It is also a distance-based geometric tool.

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