Fundamentals of Vector Algebra in 2D and 3D Euclidean Space
A Euclidean vector represents a geometric entity with both magnitude (length) and direction. Vectors form the core foundation of mechanics, 3D computer graphics, aerodynamics, and robotics.
Key Vector Equations
| Operation | Formula | Output Type |
|---|---|---|
| Magnitude ($|\vec{u}|$) | $|\vec{u}| = \sqrt{u_x^2 + u_y^2 + u_z^2}$ | Scalar |
| Dot Product ($\vec{u} \cdot \vec{v}$) | $\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z = |\vec{u}||\vec{v}|\cos\theta$ | Scalar |
| Cross Product ($\vec{u} \times \vec{v}$) | $\langle u_y v_z - u_z v_y,\ u_z v_x - u_x v_z,\ u_x v_y - u_y v_x \rangle$ | Vector (3D only) |
| Unit Vector ($\hat{u}$) | $\hat{u} = \frac{\vec{u}}{|\vec{u}|}$ | Vector (Length = 1) |