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 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).
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