The dot product takes two vectors and produces a number. For vectors in the plane, multiply matching components, then add the results.

Let $\mathbf{u}=(3,2)$ and $\mathbf{v}=(4,0)$. Multiply the horizontal components, multiply the vertical components, and add:

The result is 12, not another vector.

The same dot product can be described using the lengths of the vectors and the angle between them:

Adding and scaling say nothing about length or angle. One more operation, multiplying two vectors position by position and adding up, brings both back: it gives the length of a vector, the angle between two, and above all a test for being at right angles.

A shadow cast straight down onto the floor is longest when the stick lies flat and vanishes when it stands upright. The dot product measures how much of one vector lies along another in the same way.

$\mathbf{u}=(2,1)$ and $\mathbf{v}=(-1,2)$ meet at a right angle: $\mathbf{u}\cdot\mathbf{v}=-2+2=0$. Walked tip to tail they make a right triangle whose hypotenuse is , and .

The dot product of two vectors of $\mathbb{R}^n$ multiplies them position by position and adds; written with matrices it is $\mathbf{u}^{T}\mathbf{v}$. The length of $\mathbf{v}$ is $\mathrm{\parallel }\mathbf{v}$ and the distance between and is . The angle between non-zero vectors is defined by , a number between and by the theorem below. Vectors are orthogonal when .

• $\mathbf{u}\cdot\mathbf{v}=\mathbf{v}\cdot\mathbf{u}$; $(c\mathbf{u}+\mathbf{w})\cdot\mathbf{v}=c\,\mathbf{u}\cdot\mathbf{v}+\mathbf{w}\cdot\mathbf{v}$; unless .

If $\mathbf{v}=\mathbf{0}$ both sides are $0$. Otherwise $\|\mathbf{u}-t\mathbf{v}\|^2$ is never negative, whatever is, and expanding it gives a quadratic in . At the where the quadratic is smallest, being non-negative says exactly . Only the three rules of the dot product were used.

If $\mathbf{v}$ is $\mathbf{0}$, both sides are $0$. From now on $\mathbf{v}\neq\mathbf{0}$.

A squared length is never negative, for every real $t$; expand it with the rules.

Take the $t$ where the quadratic is smallest.

Substitute and simplify.

Multiply by $\|\mathbf{v}\|^2$ and take square roots.

The dot product pairs the vectors position by position, multiplies each pair and adds. The result is a number, not a vector.

The length of a vector is the square root of its dot product with itself: Pythagoras in every dimension.

The cosine of the angle between two non-zero vectors is their dot product divided by both lengths.

• $\mathbf{u}\cdot\mathbf{v}=u_1v_1+\cdots+u_nv_n$; .

Great work! You now know how to:

• compute dot products, lengths, distances and angles;
• test two vectors for being orthogonal;
• prove and use the Cauchy–Schwarz inequality;
• use Pythagoras for orthogonal vectors.