Vectors and Forces

Vector Components

splitting a force into across and up

A force along a slope splits into a horizontal part and a vertical part using a 3-4-5 triangle, with no trigonometry.

Example

A force pointing along a slope can be split into a horizontal part and a vertical part using a 3-4-5 triangle, no trigonometry needed.

highlighted = computed this step

A pull along a slope

A rope pulls with a force of 5 newtons along a slope that rises 3 for every 4 across. We want the across part and the up part separately.

F=5 N,slope 3:4F = 5\ \text{N}, \quad \text{slope } 3:4
A force along a rising slopeA single force arrow pointing up and to the right along a steadily rising slope.OF

The 3-4-5 triangle

The slope 3 up and 4 across has length 5, because 3 and 4 are the legs of a right triangle, so the across fraction is 4 over 5 and the up fraction is 3 over 5.

32+42=9+16=25=523^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}

The across part

Multiply the force by the across fraction: 5 times 4 over 5 is 4 newtons.

Fx=5 N⋅45=4 NF_x = 5\ \text{N} \cdot \frac{4}{5} = \hl{4}\ \text{N}

The up part

Multiply the force by the up fraction: 5 times 3 over 5 is 3 newtons.

Fy=5 N⋅35=3 NF_y = 5\ \text{N} \cdot \frac{3}{5} = \hl{3}\ \text{N}
The force split into across and up partsThe original force arrow with its horizontal across part and vertical up part drawn from the same point.OFFxFy
mechanics A 3-4-5 slope keeps the components exact whole numbers, so the across and up parts can be read off directly.