Dot Product of Two Vectors

The dot product, or scalar product, is a fundamental operation in vector algebra that combines two vectors to yield a scalar quantity. It is particularly valuable in physical applications, such as calculating work, and in mathematical contexts, such as determining vector projections and direction cosines.

For vectors a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, the dot product is defined as:

\begin{equation*}\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\end{equation*}

This operation is commutative and distributive over vector addition and is compatible with scalar multiplication. If vectors a and b are orthogonal, then their dot product equals zero. When vectors have equal direction, the dot product is maximized for their magnitudes.

The dot product also has a geometric definition involving the angle θ between two vectors:

\begin{equation*}\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\end{equation*}

This formulation determines the angle between vectors or checks for orthogonality (θ = 90° implies a·b = 0).

The angles between a vector and the positive coordinate axes are known as direction angles. If α, β, and γ are the angles between vector a and the x-, y-, and z-axes, respectively, the direction cosines are:

\begin{equation*}\cos \alpha = \frac{a_1}{|\mathbf{a}|}, \quad \cos \beta = \frac{a_2}{|\mathbf{a}|}, \quad \cos \gamma = \frac{a_3}{|\mathbf{a}|}\end{equation*}

These satisfy the identity:

\begin{equation*}\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\end{equation*}

The scalar projection of b onto a (compₐ b) and the vector projection (projₐ b) extend compₐ b by scaling the unit vector in the direction of a:

\begin{align*}\text{comp}_{\mathbf{a}} \mathbf{b} &= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}\\\text{proj}_{\mathbf{a}} \mathbf{b}& = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a}\end{align*}

These projections describe how much of b aligns with a, with physical significance in mechanics and electromagnetism.