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Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors in three-dimensional space, or the central angle subtended by the radii through two points on a sphere.
The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Thus, a ⋅ b = | a | | b | cos θ {\displaystyle \mathbf {a} \cdot \mathbf {b} =|\mathbf {a} |\,|\mathbf {b} |\cos \theta } Alternatively, it is defined as the product of the projection of ...
Relevant is the distinction between polar and axial vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors (elements of grade two). The I {\displaystyle I} here is a unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named ...
Parallel transport of a vector around a closed loop (from A to N to B and back to A) on the sphere. The angle by which it twists, , is proportional to the area inside the loop. In differential geometry, parallel transport (or parallel translation [a]) is a way of transporting geometrical data along smooth curves in a manifold.
The scalar projection is defined as [2] = ‖ ‖ = ^ where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b ...
The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel. Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors.
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself).