Dot Product

4 min read#linear-algebra

A single number, built from two vectors, that encodes the angle between them and how much one casts onto the other.

Contents

Dot Product

The dot product of two vectors multiplies them into a single number that measures how much they point the same way. Two equivalent formulas define it:

\vec a \cdot \vec b = a_x b_x + a_y b_y = \lVert \vec a\rVert\,\lVert \vec b\rVert \cos\theta.
(1)

The left form is pure arithmetic — multiply matching components and add. The right form reveals the geometry: it is the product of the two lengths times the cosine of the angle \theta between them. Setting the two equal lets you recover an angle from coordinates, which is the seed of nearly all geometry done with numbers.

Projection

The dot product is the engine of projection — dropping one vector's shadow onto another's direction. The component of \vec a along a unit vector \hat b is just \vec a \cdot \hat b, and the projected vector is

\operatorname{proj}_{\vec b}\,\vec a = \frac{\vec a \cdot \vec b}{\vec b \cdot \vec b}\,\vec b.

Drag the tip of \vec a below and watch its shadow slide along \vec b. The projection is longest when the vectors align and shrinks to nothing when they are perpendicular.

Drag to move vector a (blue). Its projection onto b (amber) is shown as the green shadow; the dashed line is the perpendicular dropped from a's tip.

Orthogonality

See also

Related topics

  • BasisA set of reference directions that turns a vector into a list of numbers — change the directions and the numbers change, though the vector does not.
  • VectorAn arrow with magnitude and direction — the basic object that linear algebra adds, scales, and transforms.
  • Linear TransformationA way of warping space that keeps grid lines straight, evenly spaced, and pinned to a fixed origin — exactly what a matrix encodes.
  • Fourier SeriesA periodic signal is an exact sum of harmonics — sinusoids at integer multiples of one fundamental frequency.
  • Linear AlgebraThe mathematics of vectors and the linear transformations that move them — the quiet engine beneath stability analysis, Fourier methods, and network ranking.
  • Gradient DescentThe fundamental learning algorithm — repeatedly step a little way downhill along the negative gradient of a loss until you reach a valley.

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