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.

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

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

where {{ "\\theta" | katex }} is the angle between the two vectors.

If vectors are expressed in component form:

Then the dot product is:

{{ "\\mathbf{A} \\cdot \\mathbf{B} = A_x B_x + A_y B_y" | katex }}

Given a vector with magnitude |V| and angle {{ "\\theta" | katex }} from the positive x-axis:

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)

The dot product is used in: