Direction Cosines and Direction Ratios of a Line

If a directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively, then these are called the direction angles of the line.

The cosines of these angles are called the direction cosines of the line.

So, the direction cosines are written as (l, m, n).

Any three numbers proportional to the direction cosines of a line are called the direction ratios of the line.

If (a, b, c) are direction ratios, then:

If (a, b, c) are the direction ratios of a line, then the corresponding direction cosines (l, m, n) are:

The sign (\[\pm\]) depends on the chosen direction of the line.

If a line passes through two points \[P(x_1, y_1, z_1)\] and \[Q(x_2, y_2, z_2)\], one set of direction ratios is:

Consequently, the direction cosines (l, m, n) are given by:

where \[PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] is the distance between the two points.

Find the direction cosines of the line passing through the two points (– 2, 4, – 5) and (1, 2, 3).

Solution: We know the direction cosines of the line passing through two points P(x₁, y₁, z1 ) and Q(x₂ , y₂, z₂ ) are given by, 

\[\frac{x_2-x_1}{\mathrm{PQ}},\frac{y_2-y_1}{\mathrm{PQ}},\frac{z_2-z_1}{\mathrm{PQ}}\]

where \[\mathrm{PQ}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}\]

Here, P is (– 2, 4, – 5) and Q is (1, 2, 3).

So \[\mathrm{PQ}=\sqrt{\left(1-\left(-2\right)\right)^{2}+\left(2-4\right)^{2}+\left(3-\left(-5\right)\right)^{2}}=\sqrt{77}\]

Thus, the direction cosines of the line joining two points are

\[\frac{3}{\sqrt{77}},\frac{-2}{\sqrt{77}},\frac{8}{\sqrt{77}}\]

• Direction Cosines (DCs) of a line: \[(l, m, n) = (\cos \alpha, \cos \beta, \cos \gamma)\].

• Main identity: \[l^2 + m^2 + n^2 = 1\].

• Direction Ratios (DRs): Are proportional to DCs.

• Relation between DRs and DCs: If DRs are (a, b, c), then DCs are proportional to (a, b, c) divided by \[\sqrt{a^2 + b^2 + c^2}\].

• DRs for two points: For points \[P(x_1, y_1, z_1)\] and \[Q(x_2, y_2, z_2)\], DRs are \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].