Vector projection

The vector projection, also known as the vector component or vector resolution, of a vector $a$ on (or onto) a non-zero 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 $\operatorname {proj} _{\mathbf {b} }\mathbf {a}$ 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 $\operatorname {oproj} _{\mathbf {b} }\mathbf {a}$ or $a ⊥ b$ ), is the orthogonal projection of $a$ onto the plane (or, in general, hyperplane) that is orthogonal to $b$ . Since both $\operatorname {proj} _{\mathbf {b} }\mathbf {a}$ and $\operatorname {oproj} _{\mathbf {b} }\mathbf {a}$ are vectors, and their sum is equal to $a$ , the rejection of $a$ from $b$ is given by: $\operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .$

Projection of $a$ on $b$ (a₁), and rejection of $a$ from $b$ (a₂)

When $90° < θ \leq 180°$ , $a 1$ has an opposite direction with respect to $b$ .

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 $\mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.$

The projection of $a$ onto $b$ can be decomposed into a direction and a scalar magnitude by writing it as $\mathbf {a} _{1}=a_{1}\mathbf {\hat {b}}$ where $a_{1}$ 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 $a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}}$ 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 $\mathbf {a}$ and $\mathbf {b}$ as: $\operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.$