Definition

Dot product

A \cdot B = ∣ A ∣∣ B ∣ cos θ = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}

Statement

A scalar equal to the projection of one vector onto the other, scaled by their magnitudes.

Symbols and units

$A, B$	vectors being dotted	—
$A \cdot B$	dot product	—
$θ$	angle between the vectors	degrees
$A_{x}, B_{x}, A_{y}, B_{y}, A_{z}, B_{z}$	Cartesian components of A and B	—

When it applies

Any two vectors; the angle θ is taken between 0 and 180°.

Don't use when

Does not apply when a direction-only result is needed — use the cross product for a perpendicular vector, not a scalar.

Derivation

Geometric definition, using the angle θ between the vectors:

A \cdot B = ∣ A ∣∣ B ∣ cos θ

Equivalent component form: cross terms between different, mutually perpendicular axes vanish, leaving:

A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}

Worked examples

Angle between two vectors

A = (3, 4), B = (4, 3):

A \cdot B = 12 + 12 = 24, ∣ A ∣ = ∣ B ∣ = 5

cos θ = \frac{24}{25} = 0.96 \Rightarrow θ \approx 16. 3^{\circ}

Watch out for

The dot product is zero exactly when the vectors are perpendicular — that is the standard test for orthogonality.

Connects to