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Illustration showing how to find the angle between vectors using the dot product Calculating bond angles of a symmetrical tetrahedral molecular geometry using a dot product. In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow.
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 a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. It coincides with the length ‖c‖ of the vector projection if the angle is smaller than 90°. More exactly: a 1 = ‖a 1 ‖ if 0° ≤ θ ≤ 90°, a 1 = −‖a 1 ‖ if 90° < θ ≤ 180°.
The angle between two term frequency vectors cannot be greater than 90°. If the attribute vectors are normalized by subtracting the vector means (e.g., ¯), the measure is called the centered cosine similarity and is equivalent to the Pearson correlation coefficient. For an example of centering,
Equals subtracted from equals are equal. When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
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).
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so (3) to its Lie group SO(3) .