Vector Dot Product

The dot product (also called scalar product) is a fundamental operation in vector mathematics that produces a scalar value from two vectors.

Definition

For two vectors A and B, the dot product is defined as:

$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)$

where $\theta$ is the angle between the two vectors.

Component Form

If vectors are expressed in component form:

• A = (A_x, A_y)
• B = (B_x, B_y)

Then the dot product is:

$\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y$

Converting from Polar to Cartesian

Given a vector with magnitude |V| and angle $\theta$ from the positive x-axis:

• V_x = |V| cos($\theta$)
• V_y = |V| sin($\theta$)

Properties of Dot Product

• Commutative: A · B = B · A
• Distributive: A · (B + C) = A · B + A · C
• Scalar multiplication: (kA) · B = k(A · B)
• Self dot product: A · A = |A|²

Geometric Interpretation

The dot product represents:

• The projection of one vector onto another, multiplied by the magnitude of the second vector
• A measure of how much two vectors point in the same direction
• If A · B > 0: vectors point in generally the same direction (acute angle)
• If A · B = 0: vectors are perpendicular (90° angle)
• If A · B < 0: vectors point in generally opposite directions (obtuse angle)

Applications

The dot product is used in:

• Physics: Calculating work (W = F · d)
• Computer Graphics: Lighting calculations and determining visibility
• Engineering: Stress analysis and structural mechanics
• Mathematics: Finding angles between vectors and projections