Vector notation using components

In physics we describe directional quantities (position, velocity, acceleration, force) using vectors. A vector in the plane is written as a column of two components:

$\vec{S} = \begin{pmatrix} S_x \\ S_y \end{pmatrix}$

The fundamental operations are carried out component by component, which reduces every vector problem to two independent scalar problems — one along $x$, one along $y$.

Operations with vectors

• Sum: $\begin{pmatrix} a_x \\ a_y \end{pmatrix} + \begin{pmatrix} b_x \\ b_y \end{pmatrix} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix}$
• Magnitude: $\norm{\vec{v}} = \sqrt{v_x^2 + v_y^2}$
• Direction: $\theta = \arctan\!\left(\dfrac{v_y}{v_x}\right)$

Example — The rabbit's displacement

A rabbit starts from the origin, moves $4\;\text{m}$ North, then turns $30°$ clockwise and advances $2\;\text{m}$. How far is it from the origin?

The two displacements in components (axis $x$ = East, axis $y$ = North) are: $\vec{S}_1 = \begin{pmatrix} 0 \\ 4 \end{pmatrix}, \qquad \vec{S}_2 = \begin{pmatrix} 2\sin 30° \\ 2\cos 30° \end{pmatrix} = \begin{pmatrix} 1 \\ 1{,}73 \end{pmatrix}$

The total displacement is the sum component by component, and its magnitude is found via Pythagoras’ theorem: $\vec{S}_{\text{tot}} = \begin{pmatrix} 1 \\ 5{,}73 \end{pmatrix}, \qquad \norm{\vec{S}_{\text{tot}}} = \sqrt{1^2 + 5{,}73^2} = 5{,}82\;\text{m}$

The direction relative to the $x$ axis: $\beta = \arctan(5{,}73/1) = 80{,}1°$.

Vector sum of the displacements: $\vec{S}_1$ towards North, $\vec{S}_2$ at an angle, and the resultant $\vec{S}_{\text{tot}}$ (dashed).

▶ Vectors — what they really are — 3Blue1Brown

Collegamenti

Argomenti: Vettori Concetti: Componenti di un vettore Competenze: Scomposizione vettoriale · Uso della trigonometria Metodi: Scomposizione in componenti cartesiane

Esercizi collegati: Il cane nel parco · Esercizio svolto — Itinerario Imola-Lugo · Problema — Vero o falso sui vettori