Vector

For a vector v, if we want to know the unit vector v_head, we first need to calculate its dimension (there should be another glossary for vector, I'm not sure), || v ||, and then set v_head = v / ||v||. e.g. if v = ai + bj, then || v || = sqrt(a^2 + b^2). We may then notice that a unit vector v_head has a dimension of 1. [Note: v_head is a mathematical notation that cannot be expressed as text here.] Take your questions as examples.

a) v = -4i, ||v|| = sqrt( (-4)^2 + 0^2 ) = 4, v_head = -4i / 4 = -i
b) v = 3j, ||v|| = 3, v_head = j
c) v = 5i + 12j, ||v|| = 13, v_head = 5/13 * i + 12/13 * j
d) v = -3i - 4j, ||v|| = 5, v_head = -3/5 i - 4/5 j

Remarks: v = ai + bj is a vector of dimension 2. Depending on your domain, you may specify a vector of any dimension, e.g. v = a1 v1 + a2 v2 + ... + an vn, and ||v|| = sqrt(a1^2 + a2^2 + ... + an^2). The method of finding the unit vector is primarily the same.