Simple Example: Speed vs. Velocity
If, starting from home, you run down the street (call that direction positive) at a speed of 6 mph; and instantaneously turn around (to keep math simple) maintaining the constant speed of 6 mph to get back home, then you can say that your average speed for the trip was 6 mph. But the velocity in the first half of the run was +6 mph (the + sign represents a direction); and the velocity coming back home was -6 mph (opposite direction). So the average velocity of the run was (0) mph. By average velocity, I take the velocity vector, V(t), and integrate it over the total time and divide by the total time = (1/T) * integral from 0 to T of V(t)dt; and since V(t) is constant of +6 for the first half of the run, and -6 for the second half, the the constant comes out, and we get (0) mph average velocity.
Velocity and Displacement Vectors and Components:
Suppose all streets except the parkway are East, West, North, and South. Let East bet the +x axis and North be the +y axis. Suppose the parkway runs at an angle to the +x axis making a 30 degree angle with it. If you run at a constant speed of 6 mph for one hour, then your speed is 6 mph and distance travelled is 6 miles. Recalling that a 30 degree triangle is a 2,1,sqrt(3) triangle, then let the 6 mile distance be the hypotenuse of the triangle. The other two edges are along the +x axis and the +y axis. At the end of one hour, you find yourself 3 miles north of the x axis, and 5.20 miles East of the y axis (i.e., 5.2 = 6*sqrt(3)/2). In terms of cartesian coordinates, you are at (x=5.2 miles, y=3 miles).
The displacement vector is the arrow from your starting point (0,0) to the end point at (5.2 miles, 3.0 miles). The displacement vector is just the finish point - start point = (5.2,3.0) - (0,0) = (5.2,3.0) miles.
The average velocity vector is just the displacement vector divided by the duration = (5.2 miles, 3.0 miles)/(one hour) = (5.2 mph, 3.0 mph). And since we said that you were running at a constant speed in a straight line along the parkway, you can say that at any point along the parkway, your velocity was (5.2 mph, 3.0 mph); i.e., at any given moment, you were running 5.2 mph East and 3.0 mph North.
