Question

The unit vector along i cap +j cap is

Answer 1

A unit vector along a vector a is a vector whose direction is the same as that of the vector a but

Magnitude 1. It is some times denoted as a^. Vectors infact are elements of a vector- space in which vector addition and multiplication of a vector by a scalar are defined. So let say that a scalar μ exists such that

|μ a | = 1. Taking a = i + j , we see that μ^2 +μ^2 =1,therefore μ =1/sqrt(2) or μ = -1/sqrt (2).Thus the required unit vector along the vector i +j is

(1/sqrt (2)) ( i +j ) or (-1/sqrt (2)) (i +j).But

(-1/sqrt (2)) ( i +j ) is unit vector along the given vector but of opposite direction. Hence the required unit vector along the given vector is

(1/sqrt (2)) ( i +j).