Vector projection is a fundamental concept in mathematics and physics that plays a crucial role in understanding the relationships between vectors and solving real-world problems. 

In this mathematics article, we will delve into the concept of vector projection, explore its applications, and provide techniques to master this powerful tool.

What is Vector Projection?

Vector projection helps us understand how much one vector points in the direction of another. You can think of it like a shadow: if you shine a light on one vector, its shadow on another vector shows how much they line up.

To find the projection of one vector a onto another vector b, we use a formula that involves the angle between them. Instead of using complicated terms, just remember:

• It tells us how much of vector a goes in the direction of vector b.
• The result is another vector (not just a number), which has a certain length and points in the same direction as b.
• This is useful in physics, engineering, and math when we want to break forces or movements into simpler parts.

Imagine a block being pulled along a ramp inclined at an angle. We have two vectors involved in this scenario: the gravitational force vector acting vertically downwards and the force vector applied to pull the block along the ramp.

The vector projection of the applied force onto the direction of the ramp determines how much of the gravitational force contributes to the block's motion along the ramp.It informs us of the effective force acting along the direction of the ramp, considering the sloping surface.

By computing the vector projection, we can find out the precise force that is responsible for the motion of the block along the ramp and thus analyze its acceleration, speed, and similar factors. 

Vector Projection Formula

In vector algebra, when we want to find the projection of a vector \(a\) on a vector \(b\), we use a formula. This formula involves multiplying the dot product of vector \(a\) and vector \(b\) by the reciprocal of the magnitude of vector \(b\). The dot product gives us a scalar value, and the magnitude of the vector \(b\) is also a scalar value. As a result, the magnitude and direction of the projection vector are both scalar values and align with the vector \(b\).

Projection of vector \(a\) on \(b\) formula \(= \frac{\vec{a}.\vec{b}}{|\vec{b}|}\)

Derivation of Vector Projection Formula

To better understand and derive the formula for projecting one vector onto another, let's consider two vectors: \(OA\) (represented as \(\vec{a}\)) and \(OB\) (represented as \(\vec{b}\)). The angle between \(\vec{a}\) and \(\vec{b}\) is denoted as \(\theta\). The projection vector represents the component of \(\vec{a}\) that aligns with \(\vec{b}\). To visualize this, draw a perpendicular line \(AL\) from point \(A\) to line \(OB\).

From the right triangle \(OAL\), \(\cos(\theta) = \frac{OL}{OA}\)

\(OL = OA \cos(\theta)\)

\(\Rightarrow\) \(OL = |\vec{a}|\cos(\theta)\)

\(OL\) represents the projection of vector \(\vec{a}\) onto vector \(\vec{b}\).

\(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos(\theta)\)

\(\Rightarrow\) \(\vec{a}\cdot\vec{b} = |\vec{b}|(|\vec{a}|\cos(\theta))\)

\(\Rightarrow\) \(\vec{a}\cdot\vec{b} = |\vec{b}|OL\)

\(\Rightarrow\) \(OL = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\)

Thus, projection vector formula of vector \(\vec{a}\) on \(\vec{b}\) \(= \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\).

Similarly, the projection of vector \(\vec{b}\) on \(\vec{a}\) \(= \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\).

How to Find the Projection of a Vector?

To find the projection of a vector onto another vector, you can follow these steps:

Step 1: Calculate the dot product.

• Multiply the corresponding components of the two vectors and sum them up.
• Example: Let's say we have vector \(\vec{A} = (3, 2)\) and vector \(\vec{B} = (1, 4)\). The dot product of \(\vec{A}\) and \(\vec{B}\) is: \((3 \times 1) + (2 \times 4) = 3 + 8 = 11\).

Step 2: Determine the magnitude of the second vector.

• Find the length or magnitude of the vector onto which you want to project.
• Example: For vector \(\vec{B} = (1, 4)\), the magnitude is found using the Pythagorean theorem: \(|B| = \sqrt{(1^2) + (4^2)} = \sqrt{1 + 16} = \sqrt{17}\).

Step 3: Divide the dot product by the magnitude.

• Divide the result from step 1 by the magnitude of the second vector.
• Example: Using the previous values, the projection of \(\vec{A}\) onto \(\vec{B}\) is: \(\frac{11}{\sqrt{17}}\).

Step 4: Simplify the projection.

• If needed, you can simplify the projection by rationalizing the denominator.
• Example: Rationalizing the denominator, the projection is approximately \(2.68\).

The resulting value represents the projection of vector \(\vec{A}\) onto vector \(\vec{B}\). It tells us the component of vector \(\vec{A}\) that aligns with the vector \(\vec{B}\) direction.

Concepts Concerning Vector Projection

In order to gain a better understanding of vector projection, we can consider the concepts that are listed below. They encompass determining the angle between two vectors and determining the dot product of two vectors based on a formula. Let us proceed to the details and understand how these concepts help in understanding vector projection.

Angle Between Two Vectors

To determine the angle between two vectors, we apply a formula that includes their dot product and magnitudes. The dot product of their components is divided by the product of their magnitudes. This will provide us with the cosine of the vectors' angle. Therefore, the formula for the angle between two vectors relies on their dot product and magnitudes.

The equation for the angle between the two vectors is as under:

\(\cos(\theta) = \frac{\vec{a}\cdot\vec{b}}{|a|\cdot|b|}\)

\(\Rightarrow\)\

\($\cos (\theta)=\frac{a_1 b_1+a_2 b_2+a_3 b_3}{\sqrt{a_1^2+a_2^2+a_3^2} \cdot \sqrt{b_1^2+b_2^2+b_3^2}}$\)

Dot Product of Two Vectors

The dot product, or scalar product, or inner product, is a mathematical operation on two vectors that results in a scalar (one number). It measures the relationship between the two vectors, namely the degree of alignment or parallelism of the vectors.

To calculate the dot product of two vectors, we multiply their corresponding components and then sum up the products. In other words, for two vectors \(\vec{A}\) and \(\vec{B}\), the dot product \((\vec{A}\cdot\vec{B})\) is given by:

\((\vec{A} \cdot \vec{B}) = A_{1} \times B_{1} + A_{2} \times B_{2} + ... + A_{n} \times B_{n}\)

Here, \(A_{1}, A_{2}, ..., A_{n}\) represent the components of vector \(\vec{A}\), and \(B_{1}, B_{2}, ..., B_{n}\) represent the components of vector \(\vec{B}\). The dot product considers the product of corresponding components and adds them together.

Applications of Vector Projection

Important applications of vector projections are listed below:

• Decomposing Vectors: Vector projection helps break down complex vectors into simpler components. By projecting a vector onto another, we can determine how much of the original vector lies in the direction of the target vector.
• Analyzing Relationships: Vector projection is useful for understanding the relationship between two vectors. It helps identify if the vectors are parallel, orthogonal (perpendicular), or have any degree of alignment.

Component Vectors: By projecting the vector, we can also obtain the component vectors of a vector. That is, we can see how much each particular direction contributes to the vector, allowing us to analyze and handle it more conveniently.

Motion and Forces: In physics, vector projection aids in the study of motion and forces. It allows us to decompose forces or velocities into components, i.e., horizontal and vertical directions, and thus facilitate proper calculation and forecast.

Geometry and Distances: Vector projection is utilized for the computation of distances and finding shortest paths. It is applied in computer graphics, engineering, and other areas that involve the measurement of distances between planes, lines, or points.

In summary, vector projection is a strong mathematical concept that finds extensive usage in mathematics, physics, and engineering. Knowing vector projection allows us to decompose complicated vectors, study vector relationships, and solve problems in real life with forces and movement.

Properties of Vector Projection

When we use the vector projection formula, a few important things depend on the angle (θ) between the two vectors. Here's what happens in different cases:

1. When θ = 90° (Right Angle):
   The projection becomes zero. This means the first vector is completely perpendicular to the second vector, so there’s no part of it pointing in the same direction.
2. When 90° < θ ≤ 180°:
   The projection will be a negative value. This means the vectors are pointing in opposite directions, like one is going forward and the other backward.
3. When 0° ≤ θ < 90°:
   The projection will be positive and points in the same direction as the second vector. This means the two vectors are going mostly in the same direction.

Vector Projection in Real-Life Situations

Vector projection is a helpful tool used in many real-world problems. It helps break down one direction into another direction, kind of like figuring out how much one path follows another. Here are some easy-to-understand examples:

GPS Navigation
Where it's used: In maps and navigation apps.
How it helps: GPS uses vector projection to find the shortest and most accurate route between two locations. It projects one location’s direction onto the Earth's surface to guide us better.

Sports Analytics
Where it's used: In games like football or basketball.
How it helps: Coaches and analysts study player movement by projecting their paths onto the field. This helps understand how fast, straight, or effective a player’s motion is.

Wind Turbine Design
Where it's used: In building wind turbines for green energy.
How it helps: Engineers use vector projection to understand how wind hits the turbine blades. This helps them set the blade angle just right to get the most power.

 Augmented Reality (AR)
Where it's used: In AR games and apps.
How it helps: Vector projection places virtual items correctly in the real world by matching their direction and position. This makes AR experiences more realistic and fun.

Vector Projection Solved Examples

1) Find the projection of vector \(\vec{A} = (3, 4)\) onto vector \(\vec{B} = (2, -1)\).

Solution)

Step 1: Calculate the dot product of vectors \(\vec{A}\) and \(\vec{B}\). 

\(\vec{A}\cdot\vec{B} = (3 \times 2) + (4 \times -1) = 6 - 4 = 2\).

Step 2: Calculate the magnitude of vector \(\vec{B}\). 

\(|B| = \sqrt{(2^2 + (-1)^2)} = \sqrt{(4 + 1)} = \sqrt{5}\).

Step 3: Calculate the projection vector using the formula:

Projection of vector \(A\) on \(B\) \(= \frac{\vec{A}.\vec{B}}{|\vec{B}|} = \frac{2}{\sqrt{5}}\) 

Therefore, the projection of vector \(\vec{A}\) onto vector \(\vec{B}\) is \(\frac{2}{\sqrt{5}}\).

2) Find the projection of vector \(\vec{X} = (-1, 2, 3)\) onto the line \(L\) represented by the direction vector \(\vec{D} = (1, -1, 2)\).

Solution)

Step 1: Calculate the dot product of vectors \(\vec{X}\) and \(\vec{D}\). 

\(\vec{X}\cdot\vec{D} = (-1 \times 1) + (2 \times -1) + (3 \times 2) = -1 - 2 + 6 = 3\).

Step 2: Calculate the magnitude of vector \(\vec{D}\).

\(|D| = \sqrt{(1^2 + (-1)^2 + 2^2)} = \sqrt{(1 + 1 + 4)} = \sqrt{6}\).

Step 3: Calculate the projection vector using the formula: 

Projection of vector \(X\) on \(D\) \(= \frac{\vec{X}.\vec{D}}{|\vec{D}|} = \frac{3}{\sqrt{6}}\).

Therefore, the projection of vector \(\vec{X}\) onto the line \(L\) is \(\frac{3}{\sqrt{6}}\).

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.