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The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. [30] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors.
Several methods for solving Wahba's problem are discussed by Markley and Mortari. This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the alternative formulation).
The decomposition or resolution [16] of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector ...
Figure 1: Parallelogram construction for adding vectors. This construction has the same result as moving F 2 so its tail coincides with the head of F 1, and taking the net force as the vector joining the tail of F 1 to the head of F 2. This procedure can be repeated to add F 3 to the resultant F 1 + F 2, and so forth.
The forces and torques acting on a rigid body can be assembled into the pair of vectors called a wrench. [3] If a system of forces and torques has a net resultant force F and a net resultant torque T, then the entire system can be replaced by a force F and an arbitrarily located couple that yields a torque of T.
As an example, the geometric product of two vectors = + = + since = and = and = , for other than and . A multivector A {\displaystyle A} may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
The short violet arrow represents the component of the gravitational force in the tangential axis, and trigonometry can be used to determine its magnitude. Thus, F = − m g sin θ = m a , so a = − g sin θ , {\displaystyle {\begin{aligned}F&=-mg\sin \theta =ma,\qquad {\text{so}}\\a&=-g\sin \theta ,\end{aligned}}} where g is the ...