Interactive diagram: vectors adding tip to tail. Two arrows, A and B, are joined tip to tail and the resultant R runs from the start of A to the end of B. Move an arrowhead or set the angle of B, and switch between adding and subtracting B.

3 + 4 can make 1, 5 or 7. It depends on the angle.

Velocities of 4 m/s and 3 m/s only add to 7 m/s when they point the same way. Turn one and the total shrinks, because a vector carries a direction as well as a size. Move either arrowhead or change the angle of B, then switch to subtract and watch B flip before it joins.

Tip-to-tail board A + B · adding
Move the white or amber arrowhead · the blue resultant R redraws from first tail to last tip
A + B = 5.0 m/s, 37° N of E
Angle of B, measured from A90°
0°90°180°270°360°
Resultant size |R|
5.0 m/s
Direction of R
37° N of E

To add vectors, draw the first arrow, then start the second arrow at the tip of the first: tip to tail. The resultant runs from the very first tail to the very last tip, and it is the one vector that does the same job as both together. Its size is almost never the sum of the sizes: 4 m/s and 3 m/s at right angles give 5 m/s by Pythagoras, and the direction comes from tan θ = opposite ÷ adjacent. A boat heading east at 4 m/s across a river flowing north at 3 m/s really moves at 5 m/s, 37° N of E, as seen from the bank. Order does not matter either: A + B and B + A finish on the same tip.

Know This

Add vectors tip to tail: the resultant runs from the first tail to the last tip, and subtracting means adding the reversed vector.