Adding Vectors

To add vectors, place them tip to tail: start the second arrow where the first one ends. The resultant runs from the start of the first arrow to the tip of the last.

Tip to tail: R runs from the start of A to the end of B.

On a line, adding vectors is adding signed numbers. Two people pull a box, one with $40\ \mathrm{N}$ to the right and one with $25\ \mathrm{N}$ to the left. With right as positive, the total force is $40 + (-25) = 15\ \mathrm{N}$ to the right.

When the two vectors are perpendicular, the resultant is the hypotenuse of a right triangle, so Pythagoras gives its size:

$$R = \sqrt{A^2 + B^2}$$

A hiker walks $4\ \mathrm{km}$ east, then $3\ \mathrm{km}$ north. Her displacement is $\sqrt{4^2 + 3^2} = 5\ \mathrm{km}$ from where she started, even though she walked $7\ \mathrm{km}$.