Vector (geometric)

A vector is an object with both a magnitude and a direction. In three-dimensional space, a vector has three real components, one for each spatial direction:

$\mathbf{A}=⟨a_x,a_y,a_z⟩=a_x\mathbf{i}+a_y\mathbf{j}+a_z\mathbf{k},$

where $\mathbf{i},\mathbf{j},\mathbf{k}$ are the standard unit vectors pointing in the $+x,+y,+z$ directions with length 1.

The most important thing to internalize: a vector is a displacement, not a position. The vector $⟨3,4,0⟩$ means “go 3 east, 4 north, 0 up” — it doesn’t say from where. You can draw an arrow for $⟨3,4,0⟩$ starting at the origin or anywhere else; as long as the length and direction match, it’s the same vector.

The one exception: a position vector $\mathbf{r}=⟨x,y,z⟩$, drawn from the origin to a point $P=(x,y,z)$, is implicitly used to mean a position.

Why vectors

Many engineering quantities need more than a single number. A force on a particle has strength and direction. A velocity has speed and direction. The electric field at a point has both. Real-number quantities — temperature, mass, voltage — are scalars (no direction); vectors are the natural object when direction matters.

Algebra

Addition (component-wise; geometrically, head-to-tail or parallelogram rule):

$\mathbf{A}+\mathbf{B}=⟨a_x+b_x,a_y+b_y,a_z+b_z⟩.$

Commutative, associative; the zero vector $0$ is the additive identity.

Scalar multiplication:

$c\mathbf{A}=⟨c{a}_x,c{a}_y,c{a}_z⟩.$

Positive $c$ stretches $\mathbf{A}$ by factor $c$, preserving direction; negative $c$ both stretches and flips.

Magnitude

By the 3D Pythagorean theorem,

$|\mathbf{A}|=\sqrt{a_x^2+a_y^2+a_z^2}.$

A common typographical convention writes $A$ (non-bold) for the magnitude of $\mathbf{A}$ (bold). So an unbolded letter labels the scalar magnitude of the corresponding bolded vector.

Unit vector

A vector with magnitude 1. To convert any nonzero $\mathbf{A}$ into a unit vector pointing the same way, divide by magnitude:

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

A clean decomposition of any vector: $\mathbf{A}=|\mathbf{A}|\hat{\mathbf{a}}$ — separating “how much” (a scalar) from “in what direction” (a unit vector). The same idea is exploited in polar form for complex numbers.

In coordinates other than Cartesian

In Cylindrical coordinates, a vector is written in components along $\hat{\mathbf{r}},\hat{\mathbf{\varphi }},\hat{\mathbf{z}}$. In Spherical coordinates, $\hat{\mathbf{r}},\hat{\mathbf{\theta }},\hat{\mathbf{\varphi }}$. Unlike Cartesian unit vectors, these are position-dependent — $\hat{\mathbf{\varphi }}$ at $(\rho ,{\varphi }_1,z)$ differs from $\hat{\mathbf{\varphi }}$ at $(\rho ,{\varphi }_2,z)$. Watch out when differentiating expressions in cylindrical/spherical components.

Two products

Vectors admit two distinct multiplications: the Dot product (scalar-valued, geometric meaning: “how much of one vector along another”) and the Cross product (vector-valued, specific to 3D, geometric meaning: area of the parallelogram with a perpendicular direction). These are the building blocks for all of vector calculus.