Notes related to the concept of Vectors

Speed is just a number with units of length / time (university physics students:  that makes it a scalar quantity).  Speed is equal to the magnitude of the velocity, which is a vector quantity.  In a sketch showing velocity the length of the vector represents the speed, and the direction of the vector represents the direction of motion.

Given a vector A in a plane, you can choose how to orient your x and y axes.  Then A_x is the component of the vector in the x direction, and A_y is the component in the y direction.  We will be doing some sketching exercises shortly, which should begin to clarify how this scheme is used.

A displacement vector is a vector from one point to another.  Its magnitude is the distance between the points, and its direction is the direction from the first point to the second.  For example the vector from (3, 6) to (9, 14) corresponds to the hypotenuse of a right triangle with legs 9 - 3 = 5 in the x direction and 14 - 6 = 8 in the y direction.  Its magnitude is sqrt( 5^2 + 8^2 ) = sqrt(89).  Its direction is arcTangent( 8 / 5 ).  We will study this more specifically soon.

If you know only the magnitudes of two displacement vectors S and T , you can't find their difference, which depends not only on their magnitudes but on the angle between them, and on their directions.

Two vectors whose x and y components are equal and opposite will have the same tangent.  If you know the sign of x you can determine whether the vector is in the first or fourth quadrant (positive x) or the second or third (negative x).

This note is relevant mostly to those taking multivariable calculus.  It goes beyond the scope of the University Physics course: 

If you are moving along a circle, then your velocity vector is tangent to the circle. 

If you are moving at a constant speed on the circle, then your acceleration vector is centripetal--i.e., it points toward the center of the circle, in the direction perpendicular to the velocity. 

If you are speeding up or slowing down as you move on the circle, your acceleration still has its centripetal component, but it also has a tangential component, a component either in the direction or opposite the direction of the velocity. 

The cross product of the acceleration and velocity vectors is perpendicular to the circle, and this defines the orientation of the plane of the circle.  If you want to move to a circle within a different plane, you're going to have to accelerate in the direction perpendicular to that plane. 

For any smooth motion in 3 dimensions we define three vectors that follow the moving point:  the unit tangent vector (in the direction of the velocity), the unit normal vector (perpendicular to the velocity, in the direction of the component of the acceleration which is perpendicular to the velocity), and the unit binormal vector which is perpendicular to both (in the direction of the cross product).