Direction Ratios

Let l, m, n be direction cosines of a line and a, b, c be three numbers such that l/a = m/b = 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

⅔/2 = -⅔/-2 = ⅓/1;

⅔/-2 = -⅔/2 = ⅓/-1;

⅔/4 = -⅔/-4 = ⅓/2.

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

± a/ √ (a² + b² + c²), ± b/ √ (a² + b² + c²), ± c/ √ (a² + b² + c²)

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

3/ √ [(3)² + (-4)² + (12)²], ± b/ √ [(3)² + (-4)² + (12)²], ± c/ √ [(3)² + (-4)² + (12)²]

Or 3/13, -4/13, 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 + m₁m₂ + 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

l₁/l₂ = m₁/m₂ = n₁/n₂

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 θ = (a₁a₂ + b₁b₂ + c₁c₂)/ √ (a₁² + b₁² + c₁²) √ (a₂² + b₂² + c₂²)

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 (a² ± b² ± c²/ a² + b² + c²).