This lecture extends vector arithmetic from two-dimensional to three-dimensional Cartesian space, defining a vector A as A1**i** + A2**j** + A3**k**, where the components correspond to coordinates of the terminal point, and establishing that vector magnitude is the square root of the sum of the squared components (a direct extension of the Pythagorean theorem/distance formula). The central theoretical point is structural invariance: vector addition, scalar multiplication, and subtraction are defined component-wise identically regardless of dimension (1D, 2D, 3D, and by extension n-dimensional space), because these operations are geometrically defined independent of any coordinate system. This belongs to the domain of vector algebra/analytic geometry within calculus, and the concept sets up the broader theme that algebraic structure (recipes) is coordinate-system-independent while computational convenience is not—illustrated by contrasting Cartesian coordinates with polar coordinates, where vector subtraction does not correspond to simple component-wise subtraction of magnitude and angle (requiring the law of cosines instead).
This lecture extends vector arithmetic from two-dimensional to three-dimensional Cartesian space, defining a vector A as A1**i** + A2**j** + A3**k**, where the components correspond to coordinates of…