Vectors

Definition

A vector is a quantity with both magnitude and direction. Vectors can be added to one another and multiplied by scalars (real numbers). A Cartesian coordinate system assigns three mutually perpendicular axes ($x$, $y$, $z$) to three-dimensional space; placing the tail of a vector at the origin, its head points to a coordinate triple that gives the vector’s components.

Intuition

Think of a vector as an arrow: its length is the magnitude and its pointing direction captures the direction. The same arrow can be described abstractly (magnitude + direction) or concretely via components measured along chosen axes. Changing axes changes the numbers but not the arrow itself.

Formal Description

Standard unit vectors $\mathbf{i},\mathbf{j},\mathbf{k}$ point along the positive $x$-, $y$-, $z$-axes respectively, each with length 1. A vector $\mathbf{A}$ with components $(A_1,A_2,A_3)$ is written

$$\mathbf{A}=A_1\mathbf{i}+A_2\mathbf{j}+A_3\mathbf{k}.$$

Length (independent of coordinate orientation):

$$|\mathbf{A}|=\sqrt{A_1^2+A_2^2+A_3^2}.$$

Vector addition is commutative and associative:

$$\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A},(\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}+(\mathbf{B}+\mathbf{C}).$$

Component-wise operations (with $\mathbf{B}=B_1\mathbf{i}+B_2\mathbf{j}+B_3\mathbf{k}$):

$$\mathbf{A}+\mathbf{B}=(A_1+B_1)\mathbf{i}+(A_2+B_2)\mathbf{j}+(A_3+B_3)\mathbf{k},c\mathbf{A}=c{A}_1\mathbf{i}+c{A}_2\mathbf{j}+c{A}_3\mathbf{k}.$$

Scalar multiplication is distributive:

$$k(\mathbf{A}+\mathbf{B})=k\mathbf{A}+k\mathbf{B}.$$

Multiplying by a positive scalar changes the length but not the direction of a vector.

Geometric interpretation: Vector addition is represented by placing the tail of one vector at the head of the other. For $\mathbf{A}$ and $\mathbf{B}$ sharing a tail, the vector $\mathbf{B}-\mathbf{A}$ points from the head of $\mathbf{A}$ to the head of $\mathbf{B}$.

Position vector:

$$\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.$$

Displacement vector from ${\mathbf{r}}_1$ to ${\mathbf{r}}_2$:

$${\mathbf{r}}_2-{\mathbf{r}}_1=(x_2-x_1)\mathbf{i}+(y_2-y_1)\mathbf{j}+(z_2-z_1)\mathbf{k}.$$

Applications

Newton’s equation for a mass $m$ acted on by forces ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$:

$$m\mathbf{a}={\mathbf{F}}_1+{\mathbf{F}}_2.$$

Cartesian components are used throughout physics, computer graphics, and machine learning (feature vectors, weight vectors).

Trade-offs

Component representation depends on the choice of axes; abstract vector properties (length, addition, scalar multiplication) are basis-independent. In higher dimensions or curved spaces, other coordinate systems may be more natural than Cartesian ones.

Links

• Position, Velocity and Acceleration — velocity and acceleration as vectors
• Trigonometric Functions — polar/Cartesian conversion via $\cos$ and $\sin$