Resolving vectors into components

5 minAS Level1.4.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can resolve a vector into two perpendicular components, including along and at right angles to a slope, and recombine components into a single vector.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$F_x$	the x-component of the force $F$

One vector, two components

Resolving is vector addition run backwards. A vector $F$ at angle $\theta$ to the horizontal can be replaced by two perpendicular vectors that add up to it:

$F_x = F\cos\theta \qquad F_y = F\sin\theta$

$F$ is the hypotenuse of a right-angled triangle; the components are the other two sides. The two components together have exactly the same effect as $F$ , so you can use them instead of it, never as well as it.

Definition

Component of a vector

The effective part of a vector in a given direction. A vector of magnitude $F$ at angle $\theta$ to a direction has a component $F\cos\theta$ in that direction and $F\sin\theta$ at right angles to it.

Worked example: Pulling a sledge

A child pulls a sledge with a rope at $30°$ above the horizontal. The tension in the rope is $40\ \text{N}$ . Find the horizontal and vertical components of the tension.

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