Dot product

The dot product is symmetric, \(u\cdot v=v\cdot u\), and linear in each slot: \((cu+w)\cdot v=c\,(u\cdot v)+w\cdot v\). Moreover \(u\cdot u=u_1^2+\dots+u_n^2\) is never negative and is zero only when \(u=0\). With these rules it behaves like an ordinary product, except that the result is a scalar rather than a vector.

A guiding example: \((1,2,3)\cdot(4,-5,6)=4-10+18=12\). In the plane, \(u\cdot v=\|u\|\|v\|\cos\theta\), where \(\theta\) is the angle between the vectors; the sign of the product therefore tells whether they form an acute angle (positive), a right angle (zero) or an obtuse one (negative). This geometric reading extends to \(\mathbb{R}^n\) and underlies the next two concepts.

Writing vectors as columns, the dot product is the matrix product \(u^{T}v\). Each entry of a matrix product is in fact the dot product of a row with a column. A common mistake is to believe that \(u\cdot v=0\) forces one of them to be zero: it is enough that they be perpendicular, like \((1,0)\) and \((0,1)\). Nor can a factor be «cancelled»: \(u\cdot v=u\cdot w\) only says that \(v-w\) is perpendicular to \(u\).