Vector addition and subtraction: component and parallelogram methods

by Quirino Sugon Jr.

The sum and difference of vectors a = (3,4) and b = (6,0)

Suppose we choose two points on the Cartesian coordinate system: and .  If we interpret these points as tips of the rays–which we shall now call as vectors–drawn from the origin at $(0,0)$, then we write down our vectors as

The sum and difference of these two vectors is given by

The parenthesis-and-comma notation for vectors is not amenable to algebraic manipulation. To solve this problem, we introduce two unit vectors (vectors of unit length) and . The unit vector points in the direction of positive axis, while points in the direction of positive axis. Using these two unit vectors, let us rewrite Eqs. (1a) and (1b) as

Hence,

It feels natural, doesn’t it?  You add only like terms: those with are added to those with ; those with are added to those with .  If you have a bag containing 3 apples and 4 oranges and another bag containing 6 apples, then putting all these in a single bag results to 9 apples and 4 oranges.  This is an interpretation of .  What do you think is the corresponding interpretation for ?

The sum and difference of vectors a and b using parallelogram method

The apples-and-oranges interpretation, though helpful, is not really precise.  The proper interpretation is geometrical.  In order to interpret and geometrically, we need to draw them together with and (see Fig. 1).  Notice that if we construct a parallelogram with as two of the parallel sides and as the other two parallel sides, then corresponds to one diagonal and corresponds to the other diagonal.

Let us write down some rules.  To draw , connect the tail of to the tip of , then draw  as the vector  from the tail of vector to the tip of .  On the other hand, to draw , connect the tails of vectors and , then draw as the vector from the tip of to the tip of .

In general, we can let the coefficients of and to be any scalar, and we can even include a new unit vector along the positive axis.  If we do this, then

so that

Equations (6a) and (6b) are the algebraic rules for vector addition and subtraction in three dimensions.  The geometrical interpretation is still the same as that for two dimensions.