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Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector | |, and a, b, c are the direction angles of the vector v. The direction angles a, b, c are acute or obtuse angles, i.e., 0 ≤ a ≤ π, 0 ≤ b ≤ π and 0 ≤ c ≤ π, and they denote the angles formed between v and the unit basis vectors e x, e y, e z.
For example, the orientation in space of a line, line segment, or vector can be specified with only two values, for example two direction cosines. Another example is the position of a point on the Earth, often described using the orientation of a line joining it with the Earth's center, measured using the two angles of longitude and latitude.
Three line segments with the same direction. In geometry, direction, also known as spatial direction or vector direction, is the common characteristic of all rays which coincide when translated to share a common endpoint; equivalently, it is the common characteristic of vectors (such as the relative position between a pair of points) which can be made equal by scaling (by some positive scalar ...
The azimuth is the angle formed between a reference direction (in this example north) and a line from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the zenith. An azimuth (/ ˈ æ z ə m ə θ / ⓘ; from Arabic: اَلسُّمُوت, romanized: as-sumūt, lit.
The angle of the intersection with the green plane is the red plane's apparent dip in the northward direction . When measuring or describing the attitude of an inclined feature, two quantities are needed. The angle the slope descends, or dip, and the direction of descent, which can be represented by strike or dip direction. [4]
To find the angle of a rotation, once the axis of the rotation is known, select a vector v perpendicular to the axis. Then the angle of the rotation is the angle between v and Rv. A more direct method, however, is to simply calculate the trace: the sum of the diagonal elements of the rotation matrix.
The user may choose to replace the inclination angle by its complement, the elevation angle (or altitude angle), measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle.
Angles are commonly defined according to the right-hand rule. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted.