Unit vector

A unit vector is a vector with magnitude $1$. Conventionally written with a hat: $\hat{\mathbf{a}}$.

For any nonzero vector $\mathbf{A}$, the unit vector in its direction is

$\hat{\mathbf{a}}=\frac{\mathbf{A}}{|\mathbf{A}|}.$

So any vector decomposes cleanly as $\mathbf{A}=|\mathbf{A}|\hat{\mathbf{a}}$ — magnitude times direction.

Standard unit vectors

In Cartesian coordinates:

$\hat{\mathbf{i}}=⟨1,0,0⟩,\hat{\mathbf{j}}=⟨0,1,0⟩,\hat{\mathbf{k}}=⟨0,0,1⟩.$

They are mutually orthogonal and obey the cyclic cross-product rule $\hat{\mathbf{i}}\times \hat{\mathbf{j}}=\hat{\mathbf{k}}$.

In Cylindrical coordinates: $\hat{\mathbf{r}},\hat{\mathbf{\varphi }},\hat{\mathbf{z}}$. In Spherical coordinates: $\hat{\mathbf{r}},\hat{\mathbf{\theta }},\hat{\mathbf{\varphi }}$. These curvilinear unit vectors are position-dependent: their direction changes from point to point. The Cartesian unit vectors are constant.

Why we care

Picking off a component. The dot product $\mathbf{A}\cdot \hat{\mathbf{u}}$ is the scalar projection of $\mathbf{A}$ onto the direction $\hat{\mathbf{u}}$ — “how much of $\mathbf{A}$ points along $\hat{\mathbf{u}}$.”

Specifying a direction without committing to a magnitude. Surface normals, tangent directions to curves, axes of rotation, polarization vectors — all stated as unit vectors so the magnitude carries other information.

Normalizing for stability. Computational algorithms (Gram–Schmidt, QR factorization, gradient descent step-direction) often normalize intermediate vectors to unit length to avoid numerical issues.

Unit tangent

For a curve parameterized by $\mathbf{r}(t)$, the Unit tangent vector is $\mathbf{T}(t)={\mathbf{r}}^{\prime }(t)/|{\mathbf{r}}^{\prime }(t)|$. It’s the heading at each point, separated from speed.

Unit normal

For an oriented surface, the unit normal $\hat{\mathbf{N}}$ specifies which side counts as “out.” It enters surface integrals and the flux integral ${\iint }_S\mathbf{F}\cdot \hat{\mathbf{N}}dS$.

Caveat: only nonzero vectors have a direction

The zero vector $0$ has no direction, so it doesn’t correspond to any unit vector. Dividing by $|0|=0$ is meaningless. Always check the vector you’re normalizing is nonzero — common source of singularities and bugs in numerical code.