Finding Orthogonal Projection

Finding Orthogonal Projection: A Simple Guide

Understanding orthogonal projection might seem daunting at first, but it's a fundamental concept with broad applications across various fields, including linear algebra, computer graphics, and machine learning. Simply put, orthogonal projection is the process of finding the "shadow" of a vector onto another vector or a subspace. Imagine shining a flashlight directly onto a wall – the light creates a shadow, which represents the projection of the object onto the wall's plane. This article will demystify this concept, guiding you through the process with clear explanations and practical examples.

1. Vectors and Subspaces: Setting the Stage

Before diving into projections, let's refresh our understanding of vectors and subspaces. A vector is a directed line segment, possessing both magnitude and direction. We often represent vectors in coordinate form (e.g., (2, 3) in two dimensions). A subspace is a subset of a vector space that is itself a vector space; it's closed under addition and scalar multiplication. For instance, a line passing through the origin is a subspace of a plane.

Imagine a vector v in a two-dimensional space and a line (a one-dimensional subspace) represented by vector u. Orthogonal projection seeks to find the component of v that lies directly on the line defined by u. This component is the orthogonal projection of v onto u.

2. The Formula for Orthogonal Projection

The mathematical formula for finding the orthogonal projection of vector v onto vector u is:

proj<sub>u</sub>v = ((v • u) / ||u||²) u

Let's break this down:

v • u: This represents the dot product of vectors v and u. The dot product measures the alignment of two vectors; a positive dot product indicates they point in similar directions, a negative dot product indicates opposite directions, and a zero dot product signifies they are orthogonal (perpendicular).

||u||²: This is the squared magnitude (length) of vector u. The magnitude is calculated as the square root of the sum of the squares of its components (e.g., for u = (a, b), ||u||² = a² + b²).

u: This is the vector onto which we are projecting. The entire expression ((v • u) / ||u||²) acts as a scalar, scaling vector u to the correct length to represent the projection.

3. A Step-by-Step Example

Let's illustrate with a concrete example. Suppose we have vector v = (3, 4) and vector u = (1, 0). We want to find the orthogonal projection of v onto u.

1. Calculate the dot product: v • u = (3 1) + (4 0) = 3

2. Calculate the squared magnitude of u: ||u||² = 1² + 0² = 1

3. Apply the formula: proj<sub>u</sub>v = (3 / 1) (1, 0) = (3, 0)

Therefore, the orthogonal projection of (3, 4) onto (1, 0) is (3, 0). This makes intuitive sense; projecting (3, 4) onto the x-axis (represented by (1, 0)) simply gives us the x-component of (3, 4).

4. Projection onto a Subspace

The concept extends to projecting onto higher-dimensional subspaces. While the formula changes slightly for subspaces, the core idea remains the same: find the component of the vector that lies within the subspace. This often involves using techniques from linear algebra such as Gram-Schmidt orthogonalization and matrix representation. However, the fundamental principle of finding the "closest" point in the subspace remains consistent.

5. Applications in Real-World Scenarios

Orthogonal projection has numerous applications:

Computer Graphics: Used for creating realistic shadows and reflections.
Machine Learning: Feature extraction and dimensionality reduction techniques often rely on projections.
Data Analysis: Used to find the best fit line or plane to a dataset (linear regression).
Physics: Resolving forces into components along specific directions.

Actionable Takeaways

Understand the concept of vector projection as finding the "shadow" of one vector onto another.
Memorize and understand the formula for orthogonal projection onto a vector.
Recognize the broader application of orthogonal projection to higher-dimensional subspaces.
Practice calculating projections with different vectors to solidify your understanding.

FAQs

1. What if the vectors are orthogonal? If vectors v and u are orthogonal, their dot product (v • u) is zero. Therefore, the projection of v onto u is the zero vector.

2. Can I project onto a zero vector? No, you cannot project onto a zero vector as the formula involves dividing by the squared magnitude of the vector, and division by zero is undefined.

3. What does it mean if the projection is longer than the original vector? This is not possible for orthogonal projection. The length of the projection will always be less than or equal to the length of the original vector.

4. How do I project onto a plane or higher dimensional subspace? This involves using more advanced linear algebra techniques such as matrix projections and basis vectors.

5. What are some resources for further learning? Look for introductory linear algebra textbooks or online courses focusing on vector spaces and linear transformations. Khan Academy and MIT OpenCourseWare are excellent free resources.

Search Results:

Orthogonal Projection Calculator: Find Your Vector Projections … Use this calculator to find the orthogonal projection of a vector U onto a vector V. Enter the components of vector U and vector V in the fields provided, with each component separated …

A.5: Inner Product and Projections - Mathematics LibreTexts 11 Sep 2022 · We say vectors →x and →y are orthogonal if they are at right angles, that is if →x, →y = 0. The vectors (1, 0, 0, 1) and (1, 2, 3, − 1) are orthogonal. So are (1, 1) and (1, − 1). …

Understanding Object Views: Orthogonal Projections Explained 3 days ago · Center of projection; Plane projection; Lines of projection; Different Types of Projections. Conics: The center of projection has a fixed position and is determined. …

Orthogonal Projection — Applied Linear Algebra - GitHub Pages Find the orthogonal projection matrix $P$ which projects onto the subspace spanned by the vectors

Orthogonal Projection Calculator: Vector Math Made Easy 15 Nov 2024 · By following these steps, you can easily compute orthogonal projections for any pair of three-dimensional vectors, making complex vector calculations simple and accessible.

Orthogonal bases and projections - Understanding Linear Algebra Use the projection formula from Proposition 6.3.15 to find $\bhat\text{,}$ the orthogonal projection of $\bvec=\fourvec92{-2}3$ onto $W\text{.}$ Find an orthonormal basis $\uvec_1$ and …

Orthogonal projection | Radiology Reference Article - Radiopaedia.org 10 Apr 2023 · Cases can appear normal in one projection and abnormal in the next. Consequently, plain radiographic imaging in the acute setting is seldom a single projection, …

linear algebra - Ways to find the orthogonal projection matrix ... 17 Dec 2017 · I'm a bit lost trying to find the projection matrix for an orthogonal projection onto a plane defined by the normal vector $n = (1, 1, 1)^T$. Then I can find the basis C of the plain $C …

6.3: Orthogonal Projection - Mathematics LibreTexts 17 Sep 2022 · Learn the basic properties of orthogonal projections as linear transformations and as matrix transformations. Recipes: orthogonal projection onto a line, orthogonal …

Orthogonal projections and Least Squares - Purdue University Our goal is to find an explicit formula for projections. From this formula the theorem can be also derived. Proposition 1. If y is the projection of x on U then y ∈ U and (x−y, u) = 0 for all u ∈ U. …

Orthogonal Projections - GeeksforGeeks 16 Oct 2021 · We need to calculate such that : Let’s take the dot product of u_1 both side. Since, this is orthogonal basis . This gives : We can generalize the above equation. Suppose {u_1, …

Subsection 6.3.2 Orthogonal Projection - gatech.edu Learn the basic properties of orthogonal projections as linear transformations and as matrix transformations. Recipes: orthogonal projection onto a line, orthogonal decomposition by …

Projection (linear algebra) - Wikipedia When has an inner product and is complete, i.e. when is a Hilbert space, the concept of orthogonality can be used. A projection on a Hilbert space is called an orthogonal projection if …

How do I find the orthogonal projection of a vector? - Socratic 19 Oct 2014 · How do I find the orthogonal projection of a vector? The orthogonal projection of → a onto → b can be found by. ⎛ ⎜ ⎜ ⎜⎝→ a ⋅ → b ∣∣ ∣→ b∣∣ ∣ ⎞ ⎟ ⎟ ⎟⎠ → b ∣∣ …

6.3: Orthogonal bases and projections - Mathematics LibreTexts 19 Jun 2024 · Use the projection formula from Proposition 6.3.15 to find $\widehat{\mathbf{b}}\text{,}$ the orthogonal projection of $\mathbf b=\fourvec92{-2}3$ onto …

Linear Algebra/Orthogonal Projection Onto a Line - Wikibooks 6 Jun 2024 · This subsection has developed a natural projection map: orthogonal projection onto a line. As suggested by the examples, it is often called for in applications.

Finding orthogonal bases - Understanding Linear Algebra There is a command to apply the projection formula: projection(b, basis) returns the orthogonal projection of b onto the subspace spanned by basis, which is a list of vectors.

Lecture 8 Orthogonal Projections - Math Find the orthogonal projection P: R3 ¡! W. Solution. We notice ﬂrst that ((1;1;1);(1;0;1)) = 2 6= 0, so this is not an orthogonal basis. Using Gram-Schmidt we get: v1 = (1;1;1) v2 = (1;0;1)¡ 2 3 …

Vector Orthogonal Projection Calculator - Symbolab Free Orthogonal projection calculator - find the vector orthogonal projection step-by-step

The formula for the orthogonal projection - University of Michigan The formula for the orthogonal projection Let V be a subspace of Rn. To nd the matrix of the orthogonal projection onto V, the way we rst discussed, takes three steps: (1) Find a basis ~v …

Master Orthogonal Projections: Key Concepts & Applications Dive into the world of orthogonal projections and discover their crucial role in linear algebra. Master key concepts, formulas, and applications to elevate your mathematical prowess.

linear algebra - Orthogonal projection of a point onto a line ... 28 Jul 2017 · Find an orthogonal projection of a point T (− 4, 5) onto a line x 3 + y − 5 = 1. The equation of your line is written in such a way that you know the x- and y-intercepts, which …

Finding Orthogonal Projection