Vector (geometric)

A vector is an object with both a magnitude and a direction. In three-dimensional space it has three real components, one per 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.

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 the arrow for $⟨3,4,0⟩$ starting at the origin or anywhere else; as long as 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 with 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). Same idea as 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, “how much of one vector along another”) and the Cross product (vector-valued, specific to 3D, gives the area of the parallelogram with a perpendicular direction).