Vector projection

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 ${\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }$ or a_∥b.

The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b (denoted ${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }$ or a_⊥b),^[1] is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both ${\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }$ and ${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }$ are vectors, and their sum is equal to a, the rejection of a from b is given by: $${\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}$$

To simplify notation, this article defines ${\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} }$ and ${\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .}$ Thus, the vector ${\displaystyle \mathbf {a} _{1}}$ is parallel to ${\displaystyle \mathbf {b} ,}$ the vector ${\displaystyle \mathbf {a} _{2}}$ is orthogonal to ${\displaystyle \mathbf {b} ,}$ and ${\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.}$

The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as ${\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }$ where ${\displaystyle a_{1}}$ is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as^[2] $${\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} }$$ 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, that is, if the angle between the vectors is more than 90 degrees.

The vector projection can be calculated using the dot product of ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$ as: $${\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}$$