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}`