Vectors Basics and Advanced for Physics

Explore triangle, parallelogram, and polygon laws of vector addition, including unit vectors and magnitude concepts, showing how to obtain sums by heads-to-tails, parallelograms, or polygon chains.

Derive the magnitude of the sum of two vectors using triangle and parallelogram methods, applying the A plus B formula with the angle between vectors and Pythagoras, using force examples.

Compute a vector’s magnitude from its components using Pythagoras in 2D and 3D. Find its direction as the angle with the positive x-axis, including two- and three-component cases.

Explore dot (scalar) and cross (vector) products for vectors, noting when each yields a scalar or a vector. Identify valid two-vector operations and invalid pairings, including scalar–vector multiplication.

Explore how the cross product yields a vector perpendicular to two given vectors, computed via determinants, with magnitude equal to the parallelogram area and direction given by the right-hand rule.

Find the angle between two vectors via the cross product, verify with the dot product, and relate cross product magnitude to parallelogram area and triangle area, plus projection concepts.