Vector projection

Article summary

The vector projection, also known as the vector component or vector resolution, of a vector on (or onto) a non-zero vector is the orthogonal projection of onto a straight line parallel to . The projection of onto is often written as \operatorname{proj}_\mathbf{b} \mathbf{a} or . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of from (denoted \operatorname{oproj}_{\mathbf{b}} \mathbf{a} or ), is the orthogonal projection of onto the plane (or, in general, hyperplane) that is orthogonal to . Since both \operatorname{proj}_{\mathbf{b}} \mathbf{a} and \operatorname{oproj}_{\mathbf{b}} \mathbf{a} are vectors, and their sum is equal to , the rejection of from is given by: \operatorname{oproj}_{\mathbf{b}} \mathbf{a} = \mathbf{a} - \operatorname{proj}_{\mathbf{b}} \mathbf{a}. To simplify notation, this article defines \mathbf{a}_1 := \operatorname{proj}_{\mathbf{b}} \mathbf{a} and \mathbf{a}_2 := \operatorname{oproj}_{\mathbf{b}} \mathbf{a}. Thus, the vector \mathbf{a}_1 is parallel to \mathbf{b}, the vector \mathbf{a}_2 is orthogonal to \mathbf{b}, and \mathbf{a} = \mathbf{a}_1 + \mathbf{a}_2.…

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