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Vector projection
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<p>The vector projection (also known as the vector component or vector resolution) of a <a href="/facts/Vector_(geometry)/bCLrdksy">vector</a> a on (or onto) a nonzero vector b is the <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projection</a> of a onto a <a href="/facts/Straight_line/gJqsr4Gj">straight line</a> 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. </p><p>The vector component or vector resolute of a <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> to b, sometimes also called the vector rejection of a <i>from</i> b (denoted oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } or a⊥b), is the orthogonal projection of a onto the <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a> (or, in general, <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a>) that is <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> to b. Since both proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } and oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } are vectors, and their sum is equal to a, the rejection of a from b is given by: oproj b a = a − proj b a . {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .} </p> <p>To simplify notation, this article defines a 1 := proj b a {\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} } and a 2 := oproj b a . {\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .} Thus, the vector a 1 {\displaystyle \mathbf {a} _{1}} is parallel to b , {\displaystyle \mathbf {b} ,} the vector a 2 {\displaystyle \mathbf {a} _{2}} is orthogonal to b , {\displaystyle \mathbf {b} ,} and a = a 1 + a 2 . {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.} </p><p>The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as a 1 = a 1 b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is a scalar, called the <i><a href="/facts/Scalar_projection/SmKRBlwm">scalar projection</a></i> of a onto b, and b̂ is the <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> in the direction of b. The scalar projection is defined as a 1 = ‖ a ‖ cos θ = a ⋅ b ^ {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} } where the operator ⋅ denotes a <a href="/facts/Dot_product/tNz8MLjT">dot product</a>, ‖a‖ is the <a href="/facts/Euclidean_norm/R2UbzmzM">length</a> of a, and <i>θ</i> is the <a href="/facts/Angle/gFUTEQYq">angle</a> 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 <a href="/facts/Opposite_vector/bCLrdksy">opposite</a> to the direction of b, that is, if the angle between the vectors is more than 90 degrees. </p><p>The vector projection can be calculated using the dot product of a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } as: proj b a = ( a ⋅ b ^ ) b ^ = a ⋅ b ‖ b ‖ b ‖ b ‖ = a ⋅ b ‖ b ‖ 2 b = a ⋅ b b ⋅ b b . {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.} </p>