Mathematical Vectors

Directed quantities, vector components, coordinate bases, and vector products in physics.

General idea of the topic

A vector is a geometric and physical quantity characterized by magnitude, direction, and sense, obeying strict coordinate rotation transformation laws. In physics, fundamental quantities such as displacement, velocity, acceleration, force, electric fields, and angular momentum are vectors. Vector algebra enables formulating physical laws in coordinate-independent intrinsic geometric forms.

This node covers vector addition/subtraction, scalar multiplication, unit base vectors ($\hat{i}, \hat{j}, \hat{k}$), dot (scalar) products ($\vec{A} \cdot \vec{B}$), cross (vector) products ($\vec{A} \times \vec{B}$), and the scalar triple product.

What you will truly understand

You will master the fundamental physical difference between the two vector multiplications: the dot product $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$ yields a coordinate-invariant scalar (such as mechanical work $W = \vec{F} \cdot \Delta \vec{r}$ or field flux), whereas the cross product $\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \,\hat{n}$ generates a vector mutually perpendicular to both (such as torque $\vec{\tau} = \vec{r} \times \vec{F}$ or the magnetic Lorentz force $\vec{F} = q \vec{v} \times \vec{B}$).

You will compute vector cross products using $3 \times 3$ determinant expansions and test vector orthogonality via the vanishing dot product condition $\vec{A} \cdot \vec{B} = 0$.

Recommended learning route

1. Vector concepts and components: magnitude, direction cosines, Cartesian components $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$, and unit direction vectors.
2. Linear vector operations: geometric parallelogram addition, component-wise addition, and scalar scaling.
3. Dot (scalar) product: algebraic and geometric definitions, orthogonal vector projections, and interior angle evaluation.
4. Cross (vector) product and triple product: right-hand rule, $3 \times 3$ determinant evaluations, parallelogram surface areas, and parallelepiped volumes.

Physical criterion you must acquire

Always remembering that the cross product is strictly anti-commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$. Swapping vector order reverses the resulting vector direction, altering the sign of torques and magnetic forces.

Connection with real problems

• Mechanics: angular momentum $\vec{L} = \vec{r} \times \vec{p}$, torque $\vec{\tau} = \vec{r} \times \vec{F}$, and gyroscopic precession.
• Electromagnetism: Lorentz force $\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})$ and Poynting electromagnetic energy flux vector $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$.
• 3D computer graphics and rendering: surface normal vectors and ray-tracing intersection algorithms.

Final goal of the topic

Master vector algebra and 3D geometry to manipulate forces, torques, fields, and spatial kinematic vectors with speed and physical accuracy.