Chain two arrows together, and you can see vector addition
Enter the coordinates of a vector — an arrow with magnitude and direction — and confirm how addition works, then use the dot product to compute the angle between two vectors yourself.
Vector addition — u+v is the arrow you get from the starting point to the tip of v, once you chain v onto the tip of u's arrow (the "triangle rule"). Below, you can type the coordinates of u and v directly as numbers, or grab the round handle at each arrow's tip with your mouse and drag it — move any one of the three arrows, and the other arrows and coordinates update instantly to match.
uvu+vv chained on
u = (uₓ, uᵧ)(3, 1)
v = (vₓ, vᵧ)(1, 3)
u + v = (uₓ+vₓ, uᵧ+vᵧ)
Dot product u·v = uₓvₓ+uᵧvᵧ tells you, as a number, "how aligned two vectors are pointing in the same direction." Drop v straight down (perpendicular) onto u's direction (the horizontal line), and the length of the resulting shadow (red arrow) is |v|cos θ, and the dot product is that value times |u|. When the dot product is 0, the two vectors are exactly perpendicular. Rotate v with the slider and find the moment the dot product hits 0 (a right angle).
uvv's shadow (projection)the perpendicular drop line