Vectors and Forces

Adding Forces

two pulls make one

Two forces at right angles combine by adding each direction on its own, then closing the right triangle for the resultant.

Example

Two forces at right angles combine by adding each direction on its own, then closing the right triangle to get the single resultant. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Two pulls at right angles

One force of 3 newtons pulls to the right and another of 4 newtons pulls straight up. What is the single force that replaces them?

Fright=3 N,Fup=4 NF_{\text{right}} = 3\ \text{N}, \quad F_{\text{up}} = 4\ \text{N}
Two forces at right anglesA rightward force arrow and an upward force arrow drawn from the same point.OF1F2

Add each direction on its own

The forces are already along the two directions, so the across total is 3 and the up total is 4.

Rx=3,Ry=4R_x = 3,\quad R_y = 4

Combine with the right triangle

The resultant closes a right triangle with legs 3 and 4: 9 plus 16 is 25, whose square root is 5 newtons.

R=32+42=25=5 NR = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = \hl{5}\ \text{N}
The two forces and their resultantThe rightward and upward force arrows with the single resultant arrow along their diagonal.OF1F2R

Scaled right triangles keep the same direction

Use the same 3 to 4 force direction. Scaling both legs scales the resultant by the same factor.

RxRyR3 N4 N5 N6 N8 N10 N9 N12 N15 N\begin{array}{c|c|c}R_x & R_y & R \\ \hline 3\ \text{N} & 4\ \text{N} & 5\ \text{N} \\ 6\ \text{N} & 8\ \text{N} & 10\ \text{N} \\ 9\ \text{N} & 12\ \text{N} & 15\ \text{N}\end{array}
mechanics A 3 and a 4 at right angles give a clean 5, so the resultant is exact.