Let l, m, n be direction cosines of a line and a, b, c be three numbers such that \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}\). Then we say that the direction ratios of the line are proportional to a, b, c.

If ⅔, – ⅔, ⅓ are direction cosines of a line, then its direction ratios are proportional to 2, -2, 1 or -2, 2, -1 or 4, -4, 2 because

\(\frac{\frac{2}{3}}{2}=\frac{\frac{-2}{3}}{-2}=\frac{\frac{1}{3}}{1}\);

\(\frac{\frac{2}{3}}{-2}=\frac{\frac{-2}{3}}{2}=\frac{\frac{1}{3}}{-1}\);

\(\frac{\frac{2}{3}}{4}=\frac{\frac{-2}{3}}{-4}=\frac{\frac{1}{3}}{2}\).

If the direction ratios of a line are proportional to a, b, c, then its direction cosines are

\(\pm \frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\), \(\pm \frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\), \(\pm \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\).

If the direction ratios of a line are proportional to 3, -4, 12, then its direction cosines are

\(\frac{3}{\sqrt{{{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}}+{{\left( 12 \right)}^{2}}}}\), \(\frac{-4}{\sqrt{{{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}}+{{\left( 12 \right)}^{2}}}}\), \(\frac{12}{\sqrt{{{\left( 3 \right)}^{2}}+{{\left( -4 \right)}^{2}}+{{\left( 12 \right)}^{2}}}}\).

Or \(\frac{3}{13}\), \(\frac{-4}{13}\), \(\frac{12}{13}\).

**Direction Ratios of the Line Segment Joining Two Points: **The direction ratios of the line segment joining two points P (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) are proportional to x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}

**Projection of a Line Segment on a given Line:** The projection of a line segment AB on a given line 1 is the length intercepted between the projections of its extremities A and B on the line.

The projection of a line segment AB on a line l is AB cos θ, where θ is the angle between AB and I

**Projection of a Line Segment on the Coordinate Axes: **The projections of a line segment AB with direction cosines l, m, n on the x-axis, y-axis and z-axis are l (AB), m (AB) and n (AB) respectively.

**Projection of a Line with given Direction Cosines:** The projections of the segment joining the points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) on a line with direction cosines l, m, n is |(x_{2 }– x_{1}) l + (y_{2} – y_{1}) m + (z_{2} – z_{1}) n|

**Angle between Two Lines in terms of their Direction Cosines:** The angle θ between lines whose direction cosines are l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} is given by Cosθ = l_{1}l_{2} + m_{1} m_{2} n_{1}n_{2}

**Condition for perpendicularity: **If the lines are perpendicular, then

θ = π/2 ⇒ Cos θ= 0 ⇒ l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

Hence, two lines having direction cosines l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are perpendicular if l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

**Condition for parallelism:** If the lines are parallel, then

θ = 0 ⇒ Cos 0° = 1 ⇒ l₁l_{2} + m_{1}m_{2} + n_{1}n_{2} = 1

Hence, the two lines having direction cosines l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are parallel if \(\frac{{{l}_{1}}}{{{l}_{2}}}\), \(\frac{{{m}_{1}}}{{{m}_{2}}}\), \(\frac{{{n}_{1}}}{{{n}_{2}}}\)

**Angle between Two Lines in terms of their Direction Ratios: **The angle θ between two lines whose direction ratios are proportional to a₁, b₁, c₁ and a₂, b₂, c₂ respectively is given by \(\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\).

Two lines direction ratios proportional to a₁, b₁, C₁ and a₂, b₂, C₂ respectively are perpendicular, if a₁a₂ + b₁b₂ + c₁c₂ = 0.

Two lines with direction ratios proportional to a_{1}, b_{1}, c_{1} and a₂, b₂, c₂ are parallel if a₁a₂ + b₁b₂ + c₁c₂.

If the edges of a rectangular parallelepiped are a, b, c; the angles between the four diagonals are given by \({{\cos }^{-1}}\left( \frac{{{a}^{2}}\pm {{b}^{2}}\pm {{c}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}} \right)\).