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This article aims to elucidate the principles of **scalar** and **vector projections**, underscoring their importance and how these concepts provide vital tools for understanding **multidimensional spaces**.

We will delve into their **mathematical** underpinnings, explore the differences between **scalar** and **vector projections**, and illustrate their **real-world implications** through various examples.

**Defining Scalar and Vector Projections**

In **mathematics**, **scalar** and **vector** **projections** help to understand the position of a point in space in relation to other points. Let’s break down the definitions of each.

**Scalar Projection**

The **scalar projection** (or **scalar component**) of a **vector A** onto a **vector B**, also known as the **dot product** of A and B, represents the **magnitude** of A that is in the **direction** of B. Essentially, it is the **length** of the segment of A that lies on the line in the direction of B. It is calculated as **|A|cos(θ)**, where **|A|** is the **magnitude** of A and θ is the **angle** between A and B.

Below, we present a generic example of scalar projection in figure-1.

Figure-1.

**Vector Projection**

The **vector projection** of a **vector A** onto a **vector B**, sometimes denoted as **proj_BA**, represents a **vector** that is in the **direction** of B with a **magnitude** equal to the **scalar projection** of A onto B.

Essentially, it is the **vector ‘shadow’** of A when ‘light’ is shone from B. It is calculated as **(A·B/|B|²) * B**, where · is the **dot product**, and |B| is the **magnitude** of B. Below, we present a generic example of vector projection in figure-2.

Figure-2.

**Properties**

**Scalar Projection**

**Commutative Property**

The **scalar projection** of vector A onto vector B is the same as the scalar projection of vector B onto vector A when the vectors are nonzero. This is because the **dot product**, which is used to calculate the scalar projection, is **commutative**.

**Scalability**

**Scalar projection** is directly proportional to the **magnitude** of the vectors. If the magnitude of either vector is scaled by a factor, the scalar projection scales by the same factor.

**Directionality**

The **sign** of the **scalar projection** gives information about the **direction**. A **positive** scalar projection means vectors A and B are in the **same direction**. A **negative** scalar projection indicates they are in **opposite directions**. A **zero** scalar projection means the vectors are **perpendicular**.

**Cosine Relationship**

The **scalar projection** is tied to the **cosine** of the angle between the two vectors. As a result, the **maximum scalar projection** occurs when the vectors are **aligned** (cosine of 0° is 1), and the **minimum** when they are **opposite** (cosine of 180° is -1).

**Vector Projection**

**Non-commutativity**

Unlike **scalar projections**, **vector projections** are not **commutative**. The **vector projection** of A onto B is not the same as the vector projection of B onto A, unless A and B are **parallel**.

**Scalability**

If you scale vector B, the vector onto which A is projected, the **vector projection** will scale by the **same factor**.

**Collinearity**

The **vector projection** of A onto B is **collinear** with B. In other words, it lies on the **same line** as B.

**Directionality**

The **vector projection** of A onto B always points in the **direction of B** if B is a **non-zero vector**. If the **scalar projection** is negative, the **vector projection** will still point in the same direction as B, but it would have indicated that A was in the opposite direction.

**Orthogonality**

The **vector** formed by subtracting the **vector projection** of A onto B from A is **orthogonal** (perpendicular) to B. This is called the **orthogonal projection** of A onto B and is a **fundamental concept** in many mathematical fields, especially in **linear algebra**.

**Exercise **

**Scalar Projections**

**Example 1**

Let **A** = [3, 4] and **B** = [1, 2]. Find the **scalar projection** of **A** onto **B**.

#### Solution

The formula for the scalar projection of **A** onto **B** is given by **A**.**B**/||**B**||. The dot product is:

**A**.**B** = (3)(1) + (4)(2)

**A**.**B** = 11

The magnitude of **B** is:

||**B**|| = √(1² + 2²)

||**B**|| = √5

Hence, the scalar projection of **A** onto **B** is 1**1/√5 = 4.9193**.

**Example 2**

Let **A** = [5, 0] and **B** = [0, 5]. Find the **scalar projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (5)(0) + (0)(5)

**A**.**B **= 0

The magnitude of **B** is:

||**B**|| = √(0² + 5²)

||**B**|| = 5

Hence, the scalar projection of **A** onto **B** is **0/5 = 0**. Since the vectors are perpendicular, the scalar projection is zero, as expected.

Figure-3.

**Example 3**

Let **A** = [-3, 2] and **B** = [4, -1]. Find the **scalar projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (-3)(4) + (2)(-1)

**A**.**B **= -14

The magnitude of **B** is:

||**B**|| = √(4² + (-1)²)

||**B**|| = √(17)

Hence, the scalar projection of **A** onto **B** is **-14/√(17) = -3.392**.

**Example 4**

Let **A** = [2, 2] and **B** = [3, -3]. Find the **scalar projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (2)(3) + (2)(-3)

**A**.**B **= 0

The magnitude of **B** is:

||**B**|| = √(3² + (-3)²)

||**B**|| = √(18)

||**B**|| = 3 * √2

Hence, the scalar projection of **A** onto **B** is **0/(3 * √2) = 0**. Again, since the vectors are perpendicular, the scalar projection is zero.

**Vector Projections**

**Example 5**

Let **A** = [1, 2] and **B** = [3, 4]. Find the **vector projection** of **A** onto **B**.

#### Solution

The formula for the vector projection of **A** onto **B** is given by:

**( A·B / ||B||² ) B**

The dot product is given by:

**A**.**B** = (1)(3) + (2)(4)

**A**.**B **= 11

The magnitude of **B** is:

||**B**|| = √(3² + 4²)

||**B**|| = 5

so ||**B**||**²** = 25

Hence, the vector projection of **A** onto **B** is **(11/25) [3, 4] = [1.32, 1.76]**.

Figure-4.

**Example 6**

Let **A** = [5, 0] and **B** = [0, 5]. Find the **vector projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (5)(0) + (0)(5)

**A**.**B **= 0

The magnitude of **B** is :

||**B**|| = √(0² + 5²)

||**B**|| = 5

so ||**B**||^2 = 25

Hence, the vector projection of **A** onto **B** is **(0/25)[0, 5] = [0, 0]**. This result reflects the fact that **A** and **B** are orthogonal.

**Example 7**

Let **A** = [-3, 2] and **B** = [4, -1]. Find the **vector projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (-3)(4) + (2)(-1)

**A**.**B** = -14

The magnitude of **B** is:

||**B**|| = √(4² + (-1)²)

||**B**|| = √17

so ||**B**||² = 17.

Hence, the vector projection of **A** onto **B** is **(-14/17)[4, -1] = [-3.29, 0.82]**.

**Example 8**

Let **A** = [2, 2] and **B** = [3, -3]. Find the **vector projection** of **A** onto **B**.

#### Solution

The dot product is given by:

**A**.**B** = (2)(3) + (2)(-3)

**A**.**B **= 0

The magnitude of **B** is:

||**B**|| = √(3² + (-3)²)

||**B**|| = √18

||**B**|| = 3 * √2

so ||**B**||² = 18.

Hence, the vector projection of **A** onto **B** is **(0/18)[3, -3] = [0, 0]**. Once again, because **A** and **B** are orthogonal, the vector projection is the zero vector.

**Applications **

**Scalar** and v**ector projections** have broad applications across a range of fields:

**Computer Science**

**Projections** are used in **computer graphics** and **game development**. When rendering **3D graphics** on a **2D screen**, **vector projections** help to create the illusion of depth. Furthermore, in **machine learning**, the concept of projection is used in dimensionality reduction techniques like **Principal Component Analysis (PCA)**, which projects data onto lower-dimensional spaces.

**Mathematics**

In **mathematics**, and more specifically **linear algebra**, **vector projections** are used in various algorithms. For example, the **Gram-Schmidt process** utilizes vector projections to orthogonally project vectors and create an **orthonormal basis**. Additionally, vector projections are employed in **least squares approximation methods**, where they help minimize the **orthogonal projection** of the error vector.

**Computer Vision and Robotics**

**Vector projections** are used in **camera calibration**, **object recognition**, and **pose estimation**. In **robotics**, projections are utilized to calculate robot movements and manipulations in **3D space**.

**Physics**

In **physics**, the **scalar projection** is often used to calculate **work done by a force**. Work is defined as the **dot product** of the force and displacement vectors, which is essentially the **scalar projection** of the force onto the displacement vector times the magnitude of displacement.

For example, if a force is applied at an **angle** to the **direction** of **motion**, only the component of the force in the direction of motion works. The** scalar projection** allows us to isolate this component.

**Computer Graphics and Game Development**

In **computer graphics**, particularly in **3D gaming**, **vector projection** plays a significant role in creating realistic motion and interactions.

For example, when you want a character to move along a surface, the motion in the direction perpendicular to the surface must be zero. This can be achieved by taking the desired **motion vector**, **projecting** it onto the **surface normal** (a vector **perpendicular** to the surface), and then subtracting that projection from the **original vector**. The result is a vector that lies entirely within the surface, creating a believable **motion** for the **character**.

**Machine Learning**

In **machine learning**, particularly in algorithms like **Principal Component Analysis (PCA)**, **projections** are used extensively. PCA works by **projecting** multidimensional data onto fewer dimensions (the principal components) in a way that preserves as much of the data’s variation as possible.

These principal components are **vectors,** and the projected data points are **scalar projections** onto these vectors. This process can help to simplify datasets, reduce noise, and identify patterns that might be less clear in the **full multidimensional space**.

**Geography**

In the field of **geography**, **vector projections** are used to portray the **3D Earth** on a **2D surface** (like a map or a computer screen). This involves **projecting geographical coordinates** (which can be thought of as points on a sphere) onto a **2D plane**.

There are many methods to do this (known as **map projections**), each with different advantages and trade-offs. For example, the **Mercator projection** preserves angles (which is useful for navigation) but distorts sizes and shapes at large scales.

**Engineering**

In **structural engineering**, the stress on a beam often needs to be resolved into components parallel and perpendicular to the beam’s axis. This is effectively **projecting** the stress vector in the relevant directions. Similarly, in **signal processing** (which is particularly important in electrical engineering), a signal is often decomposed into orthogonal components using the **Fourier transform**. This involves **projecting** the signal onto a set of basis functions, each representing a different frequency.

## Historical Significance

The concepts of **scalar** and **vector projections**, while they are now fundamental elements of **vector calculus**, are relatively modern developments in the field of **mathematics**. They are rooted in the invention and refinement of **vector analysis** during the **19th century**.

It is essential to remember that the idea of a **vector** itself wasn’t formally introduced until the mid-19th century. British physicist and mathematician **Sir William Rowan Hamilton** introduced **quaternions** in 1843, marking one of the first instances of a mathematical structure behaving like vectors as we understand them today.

Following Hamilton’s work, multiple mathematicians developed the notion of vectors. **Josiah Willard Gibbs** and **Oliver Heaviside**, working independently in the late 19th century, each developed systems of vector analysis to simplify the notation and manipulation of vector quantities in **three dimensions**. This work was mainly motivated by the desire to understand and encapsulate **James Clerk Maxwell’s equations** of electromagnetism more intuitively.

As part of these systems of vector analysis, the concepts of **dot** and **cross products** were introduced, and **scalar** and **vector projections** naturally arise from these operations. The dot product gives us a means to calculate the **scalar projection** of one vector onto another, and a simple multiplication by a unit vector provides the **vector projection**.

Despite their relatively recent historical development, these concepts have quickly become fundamental tools in a vast array of **scientific** and **engineering** disciplines, underlining their **profound utility** and power.

*All images were created with MATLAB.*