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 (x1, y1, z1) and Q (x2, y2, z2) are proportional to x2 – x1, y2 – y1, z2 – z1
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 (x1, y1, z1) and (x2, y2, z2) on a line with direction cosines l, m, n is |(x2 – x1) l + (y2 – y1) m + (z2 – z1) n|
Angle between Two Lines in terms of their Direction Cosines: The angle θ between lines whose direction cosines are l1, m1, n1 and l2, m2, n2 is given by Cosθ = l1l2 + m1 m2 n1n2
Condition for perpendicularity: If the lines are perpendicular, then
θ = π/2 ⇒ Cos θ= 0 ⇒ l1l2 + m1m2 + n1n2 = 0
Hence, two lines having direction cosines l1, m1, n1 and l2, m2, n2 are perpendicular if l1l2 + m1m2 + n1n2 = 0
Condition for parallelism: If the lines are parallel, then
θ = 0 ⇒ Cos 0° = 1 ⇒ l₁l2 + m1m2 + n1n2 = 1
Hence, the two lines having direction cosines l1, m1, n1 and l2, m2, n2 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 a1, b1, c1 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 \({{\cos }^{-1}}\left( \frac{{{a}^{2}}\pm {{b}^{2}}\pm {{c}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}} \right)\).