Vectors Basics and Advanced for Physics

Learn how to add vectors by tail-to-tail alignment, determine the angle between them, and use unit vectors i, j, k within a right-handed system, treating vectors as displacements.

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.

Determine direction of the resultant A+B by angle with A or B, using right triangle and parallelogram; illustrate with 3 N and 4 N yielding 5 N at 53°.

Learn how to resolve a vector into two perpendicular components along the x and y axes in the xy plane, using triangles and trigonometry to reconstruct the original vector.

learn how to resolve vectors into x and y components using angles with the x and y axes, drop perpendiculars, and unit vectors, then add components to obtain the resultant.

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.

Learn how resolving vectors into components simplifies adding multiple vectors and solving forces, including incline plane scenarios, by using x and y components and vector magnitude formulas.

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 the dot product, a scalar product defined by vector lengths and the angle between them, including zero for perpendicular vectors and self-dot equals length squared.

Compute the dot product of vectors given in x, y, z components. Multiply corresponding components and sum to obtain a scalar.

Learn to find the angle between vectors from their components using the dot product, with magnitude formulas, three-dimensional examples, and the perpendicular case.

Explore the cross product of two vectors, yielding a vector with magnitude |a||b|sin theta and a direction perpendicular to the vectors’ plane.

Explore cross product properties: parallel or antipodal vectors yield zero; maximum magnitude at 90 degrees; and perpendicularity to the containing plane, with determinant-based component calculations and the right-hand rule.

Learn determinants as the scalar value of a square matrix, compute 2x2 and 3x3 cases by expansion along the first row, and see how determinants underpin the cross product.

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.

Learn scalar and vector triple products and their geometric meaning in parallelepipeds. Compute A·(B×C) for volume and A×(B×C) = B(A·C) - C(A·B), plus components along and perpendicular to B.