A-10.4 Projection

There are two types of projections: a vector projection and an orthogonal (i.e., plane) projection.

Vector Projection: Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in the same $n$-dimensional vector space, the vector projection ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_{\overrightarrow{b}}(\overrightarrow{a})$ measures how much $\overrightarrow{a}$ contains the element of $\overrightarrow{b}$ (i.e., filtering $\overrightarrow{a}$ by $\overrightarrow{b}$). In the example of Figure 3a, $\overrightarrow{a}$’s projection on $\overrightarrow{b}$ is ${\overrightarrow{a}}_1$, where the length of ${\overrightarrow{a}}_1$ is geometrically $||a_1||=||a||\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃=||a||{\displaystyle \frac{a\cdot b}{||a||\cdot ||b||}}={\displaystyle \frac{a\cdot b}{||b||}}$. Let $\overrightarrow{b^{\prime }}$ be a unit vector of $\overrightarrow{b}$, that is $\overrightarrow{b^{\prime }}={\displaystyle \frac{\overrightarrow{b}}{||b||}}$. Then, ${\overrightarrow{a}}_1=||a_1||\cdot \overrightarrow{b^{\prime }}={\displaystyle \frac{a\cdot b}{||b||}}\cdot {\displaystyle \frac{\overrightarrow{b}}{||b||}}={\displaystyle \frac{a\cdot b}{||b|{|}^2}}\overrightarrow{b}$. Thus, ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_{\overrightarrow{b}}(\overrightarrow{a})={\displaystyle \frac{a\cdot b}{||b|{|}^2}}\overrightarrow{b}$.

Orthogonal Projection: Given the vector $\overrightarrow{x}$ and a set of mutually orthogonal vectors ${\overrightarrow{p}}_0,{\overrightarrow{p}}_1,\cdots ,{\overrightarrow{p}}_{n-1}$ which span the plane (or subspace) $P$, the orthogonal projection ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_P(\overrightarrow{x})$ measures how much element of plane $P$ is contained in $\overrightarrow{x}$ (i.e., filtering $\overrightarrow{x}$ by plane $P$). In the example of Figure 3b, vector $\overrightarrow{x}$’s projection on plane $P$ is shown as a red arrow, which is computed by summing the projection of $\overrightarrow{x}$ on each of the mutually orthogonal vectors ${\overrightarrow{p}}_0,{\overrightarrow{p}}_1,\cdots ,{\overrightarrow{p}}_{n-1}$ that span plane $P$. This is equivalent to ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_P(\overrightarrow{a})=\underset{i=0}{\overset{n-1}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{\mathsf{\text{𝖯𝗋𝗈𝗃}}}_{{\overrightarrow{p}}_i}(\overrightarrow{a})$. The computation of ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_P(\overrightarrow{x})$ can be thought of transforming $\overrightarrow{v}$ into a different coordinate system that expresses the vector space in terms of $n$ mutually orthogonal vectors.

Based on the definition of orthogonal projection, the following properties are derived:

Orthogonal Basis: In an $n$-dimensional vector space, any mutually orthogonal $n$ vectors in the vector space span a plane (or subspace) $P$ that is identical to the entire vector space. Further, an orthogonal projection of any vector in the vector space on $P$ is guaranteed to be a unique vector.

Non-orthogonal Basis: In an $n$-dimensional vector space, suppose some $n$ non-orthogonal vectors satisfy the following two conditions: (i) they span the entire vector space; (ii) they are linearly independent (i.e., one vector cannot be expressed as a linear combination of the other vectors). Then, the $n\times n$ matrix $P$ comprised of these $n$ vectors forms a basis of the entire vector space $V$, and the matrix-to-vector multiplication $P\overrightarrow{v}$ for each $\overrightarrow{v}$ in the vector space is guaranteed to give a unique vector. However, the formula ${\mathsf{\text{𝖯𝗋𝗈𝗃}}}_P(\overrightarrow{v})$ is not a valid geometric projection of the vector $\overrightarrow{\overrightarrow{v}}$ on $P$, because the $n$ basis vectors are non-orthogonal. Yet, the computation of $P\overrightarrow{v}$ can be thought of as uniquely transforming $\overrightarrow{v}$ into a different coordinate system that expresses the vector space in respect to $n$ non-orthogonal vectors in $P$.