Vectors: operations and components

Many physical quantities — position, velocity, acceleration, force — are not simple numbers, but also possess a direction: they are vectors. A vector $\vec{v}$ is characterised by three elements:

• magnitude $|\vec{v}|$, a positive number measuring its intensity;
• direction, the line along which it lies;
• sense, one of the two possible orientations along that line.

In the plane, a vector is conveniently represented as a pair of Cartesian components $\vec{v} = (v_x,\, v_y)$, i.e. the projections of the vector along the $x$ and $y$ axes. Components and magnitude-direction-sense are two equivalent descriptions of the same object: one passes from one to the other using trigonometry.

Key formula — Magnitude and components

Given the angle $\theta$ the vector forms with the $x$ axis: $|\vec{v}| = \sqrt{v_x^2 + v_y^2} \qquad v_x = |\vec{v}|\cos\theta \qquad v_y = |\vec{v}|\sin\theta \qquad \theta = \arctan\!\frac{v_y}{v_x}$

The great advantage of components is that operations between vectors are carried out component by component, reducing to simple operations between numbers:

• Sum: $(a_x,\, a_y) + (b_x,\, b_y) = (a_x + b_x,\; a_y + b_y)$
• Multiplication by a scalar: $k\,(v_x,\, v_y) = (k v_x,\; k v_y)$
• Opposite: $-\vec{v} = (-v_x,\; -v_y)$

In practice, to add two vectors it is almost always convenient to break them into their components, add the $x$‘s and $y$‘s separately, and recompose the result into magnitude and direction using the key formulae. This method — called the Cartesian components method — avoids graphical reasoning with the parallelogram rule and lends itself well to calculation, especially when more than two vectors are involved.

Collegamenti

Argomenti: Vettori Concetti: Componenti di un vettore Competenze: Scomposizione vettoriale Metodi: Scomposizione in componenti cartesiane

Esercizi collegati: Esercizio svolto — Itinerario Imola-Lugo · Problema — Vero o falso sui vettori · Problema — Risultante di due forze