Mathematics 3D Revision

Direction Cosines and Direction Ratios of a Line

=>As we know, If a directed line L passing through the origin makes angles α, β and γ with x, y and z-axes, respectively, called direction angles, then cosine of these angles, namely, cos α, cos β and cos γ are called direction cosines of the directed line L.

=> If we reverse the direction of L, then the direction angles are replaced by their supplements,

i.e., π −α , π − β and π − γ. Thus, the signs of the direction cosines are reversed.

color{blue}{"Direction Ratios : "} Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = λl, b=λm and c = λn, for any nonzero λ ∈ R.

"Note -" Some authors also call direction ratios as direction numbers.

● Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines (d.c’s) of the line. Then

l/a = m/b = n/c = k (say), k being a constant.

Therefore l = ak, m = bk, n = ck ... (1)

But l^2 + m^2 + n^2 = 1

Therefore k^2 (a^2 + b^2 + c^2) = 1

or  k = pm (1)/( sqrt (a^2 + b^2 + c^2 ) )

Hence, from (1), the d.c.’s of the line are

color{red}{ l = pm a/(sqrt (a^2 + b^2 + c^2 ) ) , m = pm b/(sqrt (a^2 + b^2 + c^2 ) ) , n = pm c/(sqrt (a^2 + b^2 + c^2) )}

=> where, depending on the desired sign of k, either a positive or a negative sign is to be taken for l, m and n.

=> For any line, if a, b, c are direction ratios of a line, then ka, kb, kc; k ≠ 0 is also a set of direction ratios. So, any two sets of direction ratios of a line are also proportional.

=> Also, for any line there are infinitely many sets of direction ratios.

Relation between the direction cosines of a line

=> Consider a line RS with direction cosines l, m, n. Through the origin draw a line parallel to the given line and take a point P(x, y, z) on this line. From P draw a perpendicular PA on the x-axis (Fig.).

=> Let OP = r, Then cos alpha = (OA)/(OP) =x/r . This gives x = lr.

Similarly, y = mr and z = nr

Thus x^2 + y^2 + z^2 = r^2 (l^2 + m^2 + n^2)

But x^2 + y^2 + z^2 = r^2

Hence color{blue}{l^2 + m^2 + n^2 = 1}