Scalar (Dot) Product of Vectors

Geometric Definition

For two nonzero vectors $\vec a$ and $\vec b$ with the angle between them $\theta$ (where $0\le\theta\le\pi$, measured with both vectors drawn from a common initial point), the scalar product (or dot product) is the real number

If either vector is $\vec 0$, the angle is undefined and $\vec a\cdot\vec b$ is taken to be $0$. Unlike vector addition, the dot product of two vectors is always a scalar, never a vector -- hence the name scalar product.

Geometrical Interpretation

$|\vec b|\cos\theta$ is the length of the projection of $\vec b$ onto the line of $\vec a$ (positive if $\theta$ is acute, negative if $\theta$ is obtuse, zero if $\theta=90°$). So $\vec a\cdot\vec b=|\vec a|\times(\text{projection of }\vec b\text{ on }\vec a)$: the dot product measures how much of one vector acts "along" the direction of the other, scaled by the first vector's own length.

Key Consequences of the Definition

• Perpendicularity test: since $\cos90°=0$, $\vec a\cdot\vec b=0\iff\vec a\perp\vec b$ (for nonzero $\vec a,\vec b$) -- the single most-used property of the dot product.
• Same direction ($\theta=0$): $\vec a\cdot\vec b=|\vec a||\vec b|$; in particular $\vec a\cdot\vec a=|\vec a|^2$, so $|\vec a|=\sqrt{\vec a\cdot\vec a}$.
• Opposite direction ($\theta=\pi$): $\vec a\cdot\vec b=-|\vec a||\vec b|$.
• Unit vectors along the axes: $\hat i\cdot\hat i=\hat j\cdot\hat j=\hat k\cdot\hat k=1$ and $\hat i\cdot\hat j=\hat j\cdot\hat k=\hat k\cdot\hat i=0$, since $\hat i,\hat j,\hat k$ are mutually perpendicular unit vectors.
• Commutativity: $\vec a\cdot\vec b=\vec b\cdot\vec a$.
• Distributivity over addition: $\vec a\cdot(\vec b+\vec c)=\vec a\cdot\vec b+\vec a\cdot\vec c$.

Component Formula -- Derivation

Let $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ and $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$. Using distributivity and the $\hat i,\hat j,\hat k$ products above,

Expanding, every cross term such as $a_1b_2(\hat i\cdot\hat j)$ or $a_2b_3(\hat j\cdot\hat k)$ vanishes because $\hat i\cdot\hat j=\hat j\cdot\hat k=\hat k\cdot\hat i=0$, and every matching term such as $a_1b_1(\hat i\cdot\hat i)$ survives with coefficient $1$. Only three terms remain:

Angle Between Two Vectors

Combining the geometric definition with the component formula gives the standard tool for finding the angle between two vectors: …