Vector length, squares, and dot products

Build length from square areas

An arrow going 3 units right and 4 units up forms the hypotenuse of a right triangle. Place a square on each side. Pythagoras tells us that the area of the large square equals the sum of the two smaller areas:

That 25 is a squared length. The side of a square with area 25 has length $\sqrt{25}=5$.

The same rule in any number of coordinates

For a real vector $x=(x_1,\ldots,x_n)$, add the squared coordinates, then take a square root:

The double bars mean Euclidean length. Squaring removes the signs, so an arrow going left contributes the same amount of squared length as an equally long arrow going right. The square root returns us from square units to length units.

A self dot product gives squared length

The dot product pairs corresponding entries and adds their products. Pair a vector with itself and every product becomes a square:

If $x$ is written as a column, the same scalar is $x^\mathsf{T}x$. Its transpose is a row, so the matrix product has shape $1\times1$.

Check the units

For $x=(-3,4)$, what are $x\cdot x$ and $\|x\|$?

Answer: The dot product is 25; the length is 5. Changing the horizontal direction did not change the length.