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DIRECTED LINE SEGMENT, DIRECTION ANGLE, DIRECTION COSINE, DIRECTION NUMBER

Vectors in space. In mathematics we encounter two kinds of vectors:

1) Vectors which are assumed to be located at some point P_{0}(x_{0}, y_{0}, z_{0}) in space (with their initial
point at P_{0}).

2) Vectors which are tacitly assumed to emanate from the origin of the coordinate system i.e. vectors with an initial point at (0,0,0).

In general, if no initial point is stated for a vector it is assumed to be of the second kind i.e. it is
assumed to emanate from the origin. Usually when we talk about vectors we are talking about
the second kind. If we are talking about the first kind we will try to make that clear. A directed
line segment in space is a vector of the first kind. It is a vector located at some point P_{0 } in space.

Def. Skew lines. Nonintersecting, nonparallel lines in space. Two lines are skew if and only if they do not lie in a common plane.

Def. Directed line segment. A line segment extending from some point P_{1} to another
point P_{2} in space viewed as having direction associated with it, the positive direction being from
P_{1} to P_{2}. A directed line segment P_{1}P_{2} corresponds to a vector which extends from point P_{1} to
point P_{2}.

Syn. directed line

Distance between two points in space. The distance d between any two points
P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) in space is given by

Def. Direction angles of a directed line segment. The angles α, β, and γ that a
line segment P_{1}P_{2} makes with the positive x, y and z coordinate directions respectively are called
the direction angles of the line segment. In other words, if the x-y-z coordinate system is
envisioned as translated to the point P_{1} of P_{1}P_{2} the direction angles α, β, and γ are the angles that
P_{1}P_{2} makes with the positive x, y and z axes.

Def. Direction cosines of a directed line segment. The direction cosines of a directed line segment are the cosines of the direction angles of the line segment.

Let two points P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) define directed line segment P_{1}P_{2}. Then the
direction cosines of P_{1}P_{2} are given by

where d is the distance between points P_{1} and P_{2}
(i.e. the length of P_{1}P_{2} ).

Direction cosines are not independent. When two of them are given, the third one can be found, except for sign, from the relation

We shall denote the direction cosines cos α, cos β, and cos γ of a directed line segment by λ, μ, and ν respectively. Intuitively, the direction cosines λ, μ, and ν of a directed line segment (i.e vector) correspond to the projections of its associated unit direction vector on the x, y and z axes. That associated unit direction vector of which we speak is a unit vector emanating from the origin pointing in the same direction as the directed line segment. Figure 1 shows the direction cosines λ, μ, and ν of a unit direction vector u emanating from the origin of the coordinate system. λ, μ, and ν correspond to the projections of the unit vector on the x, y and z axes. Thus the coordinates of the unit vector u are (λ, μ, ν).

Unit direction vectors. The direction of any directed line segment P_{1}P_{2} in space is
represented by a unit vector emanating from the origin of the coordinate system and pointed in
the same direction as P_{1}P_{2}. If P_{1}P_{2} has direction angles α, β, and γ then its associated unit
direction vector has direction angles α, β, and γ and has components given by (cos α, cos β, cos
γ) or , equivalently, (λ, μ, ν). Thus every directed line segment of space pointed in the same
direction as P_{1}P_{2} has the same representative unit direction vector as P_{1}P_{2}. The set of all unit
direction vectors emanating from the origin represents all possible directions of vectors, directed
line segments or lines in space. It thus serves as a representative set for all possible directions.

Def. Direction numbers of a directed line segment. Any three numbers proportional to the direction cosines of the line segment.

Let (2, 3, 5) be a vector emanating from the origin. Then its direction numbers l, m, n are given by its components 2, 3, 5.

Let points P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) define directed line segment P_{1}P_{2}. Then its direction
numbers l, m, n are given by the three numbers x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}, or any multiple of them

Angle between two directed line segments. The angle between any two directed line segments in space, whether the line segments intersect or not, is defined as the angle between their associated unit direction vectors. Thus the angle between two directed line segments corresponds to the angle between their directions. The idea of an angle between two intersecting lines is commonly understood but this definition extends that customary concept to the case of non-intersecting lines (or line segments).

Angle between two unit direction vectors. The angle θ between two unit
direction vectors (λ_{1}, μ_{1}, ν_{1}) and (λ_{2}, μ_{2}, ν_{2}) is given by

cos θ = λ_{1}λ_{2} + μ_{1}μ_{2} + ν_{1}ν_{2}

Thus if two directed line segments have direction angles α_{1}, β_{1}, γ_{1} and α_{2}, β_{2}, γ_{2 }, then the angle θ
between the line segments is given by

cos θ = cos α_{1} cos α_{2} + cos β_{1} cos β_{2} + cos γ_{1} cos γ_{2 }

or, equivalently,

cos θ = λ_{1}λ_{2} + μ_{1}μ_{2} + ν_{1}ν_{2} .

Angle between two lines with direction numbers l_{1}, m_{1}, n_{1} and l_{2}, m_{2},
n_{2 .} The acute angle θ between two lines with direction numbers l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2 }is given
by

Condition for perpendicularity of two lines. Two line segments with
directions (λ_{1}, μ_{1}, ν_{1}) and (λ_{2}, μ_{2}, ν_{2}) are perpendicular if and only if

λ_{1}λ_{2} + μ_{1}μ_{2} + ν_{1}ν_{2} = 0

Two line segments with direction numbers l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2 } are perpendicular if and only if

l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0 .

Sine of the angle between two lines. If θ is the angle between the directions (λ_{1}, μ_{1},
ν_{1}) and (λ_{2}, μ_{2}, ν_{2}), then

Direction perpendicular to two directions. If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2 } are
direction numbers of non-parallel lines, then a set of direction numbers of a line perpendicular to
these lines is

Note. A simple and easily remembered device for computing these direction numbers is as follows:

Write down the direction numbers twice and form the determinants provided by the three middle pair of adjacent columns.:

Example. Find a set of direction numbers for a line perpendicular to two lines having direction numbers 3, 4, -1 and 3, - 4, 3.

Solution. Form the array

We compute the direction numbers 8. -12, -24 .

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