Directed line segment, direction angle, direction cosine, direction number

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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 P₀(x₀, y₀, z₀) in space (with their initial point at P₀).

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 P₀ 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 P₁ to another point P₂ in space viewed as having direction associated with it, the positive direction being from P₁ to P₂. A directed line segment P₁P₂ corresponds to a vector which extends from point P₁ to point P₂.

Syn. directed line

Distance between two points in space. The distance d between any two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) in space is given by

Def. Direction angles of a directed line segment. The angles α, β, and γ that a line segment P₁P₂ 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 P₁ of P₁P₂ the direction angles α, β, and γ are the angles that P₁P₂ 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 P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) define directed line segment P₁P₂. Then the direction cosines of P₁P₂ are given by

where d is the distance between points P₁ and P₂ (i.e. the length of P₁P₂ ).

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 P₁P₂ in space is represented by a unit vector emanating from the origin of the coordinate system and pointed in the same direction as P₁P₂. If P₁P₂ 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 P₁P₂ has the same representative unit direction vector as P₁P₂. 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 P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) define directed line segment P₁P₂. Then its direction numbers l, m, n are given by the three numbers x₂-x₁, y₂-y₁, z₂-z₁, 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 (λ₁, μ₁, ν₁) and (λ₂, μ₂, ν₂) is given by

cos θ = λ₁λ₂ + μ₁μ₂ + ν₁ν₂

Thus if two directed line segments have direction angles α₁, β₁, γ₁ and α₂, β₂, γ₂, then the angle θ between the line segments is given by

cos θ = cos α₁ cos α₂ + cos β₁ cos β₂ + cos γ₁ cos γ₂

or, equivalently,

cos θ = λ₁λ₂ + μ₁μ₂ + ν₁ν₂ .

Angle between two lines with direction numbers l₁, m₁, n₁ and l₂, m₂, n_{2 .} The acute angle θ between two lines with direction numbers l₁, m₁, n₁ and l₂, m₂, n₂is given by

Condition for perpendicularity of two lines. Two line segments with directions (λ₁, μ₁, ν₁) and (λ₂, μ₂, ν₂) are perpendicular if and only if

λ₁λ₂ + μ₁μ₂ + ν₁ν₂ = 0

Two line segments with direction numbers l₁, m₁, n₁ and l₂, m₂, n₂ are perpendicular if and only if

l₁l₂ + m₁m₂ + n₁n₂ = 0 .

Sine of the angle between two lines. If θ is the angle between the directions (λ₁, μ₁, ν₁) and (λ₂, μ₂, ν₂), then

Direction perpendicular to two directions. If l₁, m₁, n₁ and l₂, m₂, n₂ 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 .