The University of Sydney - School of Mathematics and Statistics

Addition and subtraction of vectors in terms of components	Page 4 of 5

Given two vectors in Cartesian form

the sum + = is obtained by completing the parallelogram.

It can be proved that this is the same as the following calculation:

That is, the components of a sum are the sums of the components.

Notice that the parallelogram OQRP is part of a two dimensional plane sitting within three dimensional space (in a tilted way like the slanting face of the roof of a house).

Subtraction

The rule for subtraction works in exactly the same way. Writing for the vector -, the rule above gives

The subtraction is illustrated below. Recall that = - = .

An example of the use of this rule is the calculation of the Cartesian form of the position vector of a point P₂ relative to a point P₁. Suppose that in Cartesian form, = x₁i + y₁j + z₁k and = x₂i + y₂j + z₂k.

Then