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As indicated previously, a vector u is defined as a quantity possessing both a magnitude and a direction; the said magnitude is regularly denoted by ‖ u ‖, while information on the direction is often conveyed graphically – or else encompasses angles formed with the axes in some reference system. Two vectors, u and v, are said to be equal when their magnitudes are identical, i.e. ‖ u ‖ = ‖ v ‖, and also point in the same direction; however, they do not need to have the same origin.

A much more convenient way of handling vectors resorts, however, to their decomposition along the three directions of space in a typical Cartesian R³ domain, according to

(3.1)

and

(3.2)

here j_x, j_y, and j_z denote unit, orthogonal vectors of a Cartesian system, defined as

(3.3)

(3.4)

and

(3.5)

– while

(3.6)

and

(3.7)

define u and v, respectively, via their coordinates.

According to Pythagoras’ theorem,

(3.8)

and likewise

(3.9)

this is a more general form than Eq. (2.431), yet it relies on application of the aforementioned theorem twice. In fact,

(3.10)

abides to Eq. (2.431), as long as u_x and u_y denote the projections of u onto the x‐ and y‐axis, respectively, and u_xy denotes the projection of u onto the x0y plane; further application of Eq. (2.431) then supports

(3.11)

where u_z denotes the projection of u onto the z‐axis. Insertion of Eq. (3.10) transforms Eq. (3.11) to

(3.12)

that retrieves Eq. (3.8), after taking square roots of both sides – as long as ∣u_x ∣ ≡ ‖ u_x ‖, ∣u_y ∣ ≡ ‖ u_y ‖, and ∣u_z ∣ ≡ ‖ u_z ‖; a similar reasoning obviously applies to v_x, v_y, and v_z describing v . The general rules of trigonometry indicate, in turn, that angle θ_u (or θ_v) – formed with the y‐axis by the projection of u (or v) onto the y0z plane, is such that

(3.13)

and likewise

(3.14)

similar expressions can be laid out pertaining to angle ϕ_u (or ϕ_v) formed with the z‐axis by the projection of u (or v) onto the x0z plane, viz.

(3.15)

and similarly

(3.16)

Therefore, equality of u and v requires that ‖ u ‖ = ‖ v ‖ describing the same length, and θ_u = θ_v and ϕ_u = ϕ_v describing the same orientation – thus using up all three spatial degrees of freedom; upon inspection of the functional forms of Eqs. (3.8), (3.9), and (3.13)–(3.16), one concludes that the said three equalities unequivocally enforce u_x = v_x, u_y = v_y, and u_z = v_z .