2dimension

two vectors a and b are equal if they have the same direction and the same magnitude (or length). We could then write

a = b

A vector a may be multiplied by a scalar s (remember, a scalar is an ordinary number). We could write this new vector as

c = s a

This new vector c has the same direction as vector a and its magnitude is s times the magnitude of a.

We write the magnitude (or length) of a vector without boldface or without a vector over it. The magnitude of a vector is an ordinary scalar; there is no direction associated with the magnitude of a vector.

Speed is the magnitude of velocity and distance is the magnitude of displacement.

Consider two vectors A and B which we want to add. They might be displacement vectors or velocity vectors or electric field vectors -- or any vectors at all. We can add them graphically by drawing vector A and then, at the tip of vector A, drawing vector B as shown below. The sum of vectors is called the resultant. The resultant vector R, is the vector that we can draw from the beginning of A to the end of B. We can write this as

R = A + B

Vector addition is commutative. That means the order in which we add vectors does not affect the resultant. To add vectors A and B we could begin by drawing vector B. At the end of vector B we would then draw vector A. The resultant vector R is then the vector we can draw by starting at the beginning of B and finishing at the end of A. We could write this as

R = B + A

These resultants are exactly the same. That means

A + B = B + A

Sometime people will draw the vectors in twice to form a parallelogram . The resultant R is the diagonal as shown. This is referred to as vector addition by the parallelogram method.