Vector Lab

Mathematical Principles & Theorems

Vector calculus in \(\mathbb{R}^3\): \(\mathbf{u} = (u_x, u_y, u_z), \mathbf{v} = (v_x, v_y, v_z)\). Dot Product: \(\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_x v_x + u_y v_y + u_z v_z\) (scalar projection). Cross Product: \(\mathbf{u}\times\mathbf{v} = (u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x)\) (vector normal with magnitude equal to parallelogram area). Vector projection \(\text{proj}_\mathbf{v}\mathbf{u} = \frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}\). Vector fields \(\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}\) visualize flow fields.

Operating Instructions

• Enter vector components for \(\mathbf{u}\) and \(\mathbf{v}\) in the numeric parameter inputs.
• Click Compute to calculate Vector Sum, Dot Product, Cross Product, Angle \(\theta\), and Projections.
• In Vector Field mode: Enter field functions \(P(x,y), Q(x,y)\) and configure grid density with the value box.
• Rotate and zoom the 3D vector canvas to inspect geometric vector relationships.