We will call a sequence of numbers of length k a k-vector.
We define addition and subtraction among k-vectors to be termwise addition and subtraction, so that for 2-vectors we have
(a, b) + (c, d) = (a + c, b + d)
If we choose an origin O in the Euclidean plane we can describe any point in the plane by a vector whose first component is the x coordinate of the point and second is the y component, that is, by (x, y), a 2-vector.
We call the vector with i-th component value = 1 and the rest 0 the basis vector in the i-th direction. In ordinary three dimensional space the basis vectors in the x, y and z directions are denoted as i, j and k respectively. The vector (x, y, z) can also be written as xi + yj + zk.
We assume here that basis vectors are perpendicular to one another, and each has unit length.