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When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule. [59] The angle is the angle between the two vectors. In symbols, =.
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 ...
If vectors u and v have direction cosines (α u, β u, γ u) and (α v, β v, γ v) respectively, with an angle θ between them, their units vectors are ^ = + + (+ +) = + + ^ = + + (+ +) = + +. Taking the dot product of these two unit vectors yield, ^ ^ = + + = , where θ is the angle between the two unit vectors, and is also the angle between u and v.
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 ...
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.
If the cross product of two vectors is the zero vector (that is, a × b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sin θ = 0).
The normalized angle, referred to as angular distance, between any two vectors and is a formal distance metric and can be calculated from the cosine similarity. [5] The complement of the angular distance metric can then be used to define angular similarity function bounded between 0 and 1, inclusive.
The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. [18] It may be defined as the acute angle between two lines normal to the planes. The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the normal to the plane.