Length, Distance, Dot Product, and Orthogonality

Modern interactive lesson

Measure vector geometry with the dot product. Visualize lengths, distances, angles, orthogonality, and orthogonal complements.

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Big idea: dot product turns vectors into geometry

The dot product measures length, distance, angle, perpendicularity, and similarity.

$$u\cdot v=u^Tv=\sum_{i=1}^n u_iv_i.$$

$\|u\|=\sqrt{u\cdot u}$

$\operatorname{dist}(u,v)=\|u-v\|$

$\cos\theta=\dfrac{u\cdot v}{\|u\|\|v\|}$

The dot product is symmetric, distributive, and positive: $u\cdot u\ge0$.

$u\cdot v=0$ means $u$ and $v$ are orthogonal, assuming both are nonzero.

Computational meaning: dot products are the basic operation behind projections, least squares, cosine similarity, and many data algorithms.

dot · norm · distance

$u=(u_1,u_2)$

$v=(v_1,v_2)$

For nonzero vectors, the angle is determined by

$$\cos\theta=\frac{u\cdot v}{\|u\|\|v\|}.$$

$-1$ to $1$

Move the slider to rotate $v$ relative to $u=(1,0)$. The output is $\cos\theta$.

The orthogonal complement of a subspace contains all vectors perpendicular to every vector in the subspace.

$$W^\perp=\{x:x\cdot w=0\text{ for all }w\in W\}.$$

perpendicular line

Line direction $w=(a,b)$

A number measuring alignment.

Dot product zero.

Divide by length to get a unit vector.

If $u\cdot v=0$ and both vectors are nonzero, what does it mean?

Common mistake: the angle formula is not valid if either vector is zero.