The dot product returns a scalar that measures alignment, while the cross product returns a vector perpendicular to two three-dimensional vectors. Learn the formulas, geometric meaning, right-hand rule, projections, work, area, plane normals, torque, flux, scalar triple products, Lagrange’s identity, and complete worked examples.
Which direction makes a multivariable function increase fastest? Learn how the gradient packages partial derivatives, how to calculate directional derivatives with unit vectors and dot products, and why gradients cross level curves at right angles.
