Vector Projection: Definition, Formula, and Solved Examples

Vector projection refers to projecting one vector onto another, showing how much of one vector lies in the direction of the other. It helps break down vectors into components, making understanding directional relationships in physics, engineering, and computer graphics easier. The formula uses the dot product to compute this projection.

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1. Introduction to the Vector Projection:

In the world of vectors and linear algebra, vector projection plays a significant role in simplifying complex computations. It allows us to decompose a vector into two components: one that aligns with another vector and one that is perpendicular. This decomposition is highly useful in fields like physics, computer graphics, and engineering, where understanding directional relationships between vectors is crucial.

2. What is Vector Projection:

The vector projection of a vector ${\vec{a}}$ onto another vector ${\vec{b}}$ is the vector that points in the direction of ${\vec{b}}$ and represents how much of ${\vec{b}}$ aligns with ${\vec{b}}$. In simpler terms, it's the "shadow" of one vector onto another.

Mathematically, the projection of vector ${a}$ onto vector ${b}$ is denoted as:

$\space \text{Proj}_{{\vec{b}}}{\vec{a}} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{|\vec{b}|}.\hat{b} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{|\vec{b}|^2}.\vec{b} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{{\vec{b}} \cdot {\vec{b}}} {\vec{b}}$

Where:

• ${\vec{a}} \cdot {\vec{b}}$ is the dot product of vectors ${\vec{a}}$ and ${\vec{b}}$,
• ${\vec{b}} \cdot {\vec{b}}$ is the square of the magnitude of ${\vec{b}}$,
• The result is a vector in the ${\vec{b}}$ direction.
• $\hat{b}$ is the unit vector in the direction of $\vec{b}$

3. How to Find the Vector Projection:

The vector projection formula involves the dot product and magnitude of the vectors involved. Here's a step-by-step process to find the vector projection:

Step 1: Calculate the Dot Product
The dot product of two vectors ${\vec{a}}$ and ${\vec{b}}$ is given by: $\space {\vec{a}} \cdot {\vec{b}} = a_1b_1 a_2b_2 a_3b_3$

For $n$-dimensional vectors, the dot product includes all components.

Step 2: Find the Magnitude of the Vector
The magnitude (length) of vector ${\vec{b}}$ is: $\space ||{\vec{b}}|| = \sqrt{b_1^2 b_2^2 b_3^2}$

The square of the magnitude is ${\vec{b}} \cdot {\vec{b}}$.

Step 3: Apply the Projection Formula
Substitute the dot product and the magnitude into the projection formula: $\space \text{proj}_{{\vec{b}}} {\vec{a}} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{{\vec{b}} \cdot {\vec{b}}} \space {\vec{b}}$

This gives the projection of ${\vec{a}}$ onto ${\vec{b}}$, a vector that is parallel to ${\vec{b}}$.

4. Rules for Vector Projection:

There are some important rules to follow when working with vector projections:

1. Direction Matters: The projection of ${\vec{a}}$ onto ${\vec{b}}$ points in the same direction as ${\vec{b}}$ if the dot product is positive and in the opposite direction if the dot product is negative.

2. Zero Projection: If ${\vec{a}}$ is perpendicular to ${\vec{b}}$, the projection will be a zero vector, i.e., $\text{proj}_{{\vec{b}}} {\vec{a}} = 0$.

3. Orthogonal Projection: The vector projection is always orthogonal (perpendicular) to the vector difference between ${\vec{a}}$ and its projection on ${\vec{b}}$.

4. Projection onto Itself: The projection of a vector onto itself is just the vector itself: $\text{proj}_{{\vec{b}}} {\vec{b}} = {\vec{b}}$.

5. Properties of Vector Projection:

Here are some key properties of vector projection:

1. Linear Relationship: Vector projections preserve the linearity of the vectors. That is, if ${\vec{c}} = \alpha {\vec{a}} \beta {\vec{b}}$, then: ${\text{proj}_{\vec{b}} {\vec{c}} = \alpha \text{proj}_{\vec{b}} {\vec{a}} \beta \text{proj}_{\vec{b}} {\vec{b}}}$

2. Length of Projection: The magnitude of the projection of ${a}$ onto ${b}$ is given by: ${ \| \text{proj}_{\vec{b}} {\vec{a}} \| = \dfrac{ | {\vec{a}} \cdot {\vec{b}} | }{ \| {\vec{b}} \| }}$

3. Projection is Parallel: The resulting vector from the projection is always parallel to the vector onto which the projection is being made.

4. Angle Between Vectors: The closer the angle between two vectors is to 90 degrees, the smaller the projection becomes. The projection is zero if the vectors are orthogonal (90 degrees).

6. Vector Projection Solved Examples:

Question: 1. Projection of a 2D Vector
Find the projection of ${\vec{a}} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$.

Solution:
Step 1: Find the Dot Product
The dot product of ${\vec{a}}$ and ${\vec{b}}$ is: ${\vec{a}} \cdot {\vec{b}} = (3)(1) (4)(2) = 3 8 = 11$

Step 2: Find the Magnitude of ${b}$
The magnitude of ${\vec{b}}$ squared is: $\| \mathbf{\vec{b}} \|^2 = (1)^2 (2)^2 = 1 4 = 5$

Step 3: Apply the Projection Formula
Now apply the vector projection formula: $\text{proj}_{{\vec{b}}} {\vec{a}} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{{\vec{b}} \cdot {\vec{b}}} {\vec{b}} = \dfrac{11}{5} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} \dfrac{11}{5} \\ \\ \dfrac{22}{5} \end{pmatrix}$

Final Answer: $\text{proj}_{\vec{b}} {\vec{a}} = \begin{pmatrix} 2.2 \\ 4.4 \end{pmatrix}$

Question: 2. Projection of a Vector in 3D
Find the projection of ${\vec{a}} = \begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$.

Solution:
Step 1: Calculate the Dot Product
The dot product of ${\vec{a}}$ and ${\vec{b}}$ is: ${a} \cdot {\vec{b}} = (4)(1) (2)(3) (-1)(2) = 4 6 - 2 = 8$

Step 2: Calculate the Magnitude of ${b}$
The magnitude of ${\vec{b}}$ squared is: $\| {\vec{b}} \|^2 = (1)^2 (3)^2 (2)^2 = 1 9 4 = 14$

Step 3: Apply the Projection Formula
Now apply the vector projection formula: $\text{proj}_{\vec{b}} {\vec{a}} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{{\vec{b}} \cdot {\vec{b}}} {\vec{b}} = \dfrac{8}{14} {\vec{b}} = \dfrac{4}{7} \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} = \begin{pmatrix} \dfrac{4}{7} \\ \\ \dfrac{12}{7} \\ \\ \dfrac{8}{7} \end{pmatrix}$

Final Answer: $\text{proj}_{\vec{b}} {\vec{a}} = \begin{pmatrix} \dfrac{4}{7} \\ \\ \dfrac{12}{7} \\ \\ \dfrac{8}{7} \end{pmatrix}$

Question: 3. Projection of a 2D Vector onto a Perpendicular Vector
Find the projection of ${\vec{a}} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$, which are perpendicular vectors.

Solution:
Step 1: Calculate the Dot Product ${\vec{a}} \cdot {\vec{b}} = (1)(-2) (2)(1) = -2 2 = 0$

Step 2: Calculate the Magnitude of ${\vec{b}}:$ $\| \mathbf{\vec{b}} \|^2 = (-2)^2 (1)^2 = 4 1 = 5$

Step 3: Apply the Projection Formula Since the dot product is zero, the projection of ${a}$ onto ${b}$ is the zero vector:

$\text{proj}_{\vec{b}} {\vec{a}} = \dfrac{0}{5} {\vec{b}} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Final Answer: $\text{proj}_{\vec{b}} {\vec{a}} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

7. Practice Questions on Vector Projection:

Q.1: Find the projection of ${\vec{a}} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$.

Q.2: Calculate the projection of ${\vec{a}} = \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$.

Q.3: Determine the vector projection of ${\vec{a}} = \begin{pmatrix} 0 \\ 3 \\ -1 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$.

Q.4: What is the projection of ${\vec{a}} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}$ onto ${\vec{b}} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$?

8. FAQs on Vector Projection:

What is vector projection in simple terms?

Vector projection is the process of finding the component of one vector that lies in the direction of another vector. It shows how much of one vector aligns with or contributes to another vector’s direction.

How do you calculate vector projection?

The formula for the projection of vector ${\vec{a}}$ onto vector ${\vec{b}}$ is: $\text{proj}_{\vec{b}} {\vec{a}} = \dfrac{{\vec{a}} \cdot {\vec{b}}}{{\vec{b}} \cdot {\vec{b}}} {\vec{b}}$ You need to compute the dot product of ${\vec{a}}$ and ${\vec{b}}$, and divide it by the dot product of ${\vec{b}}$ with itself, then multiply the result by ${\vec{b}}$.

What is the difference between scalar and vector projection?

Scalar projection gives the magnitude (a number) of the projection of one vector onto another, whereas vector projection gives both magnitude and direction.

When is the projection of a vector zero?

The projection of a vector onto another vector is zero when the vectors are perpendicular to each other, meaning their dot product is zero.

Can vector projection be negative?

Yes, the projection of a vector can be negative when the two vectors point in opposite directions. This indicates that the projected vector is pointing in the opposite direction of the vector onto which it is projected.

What is the physical significance of vector projection?

In physics, vector projection helps break down forces into components, such as resolving a force vector into parallel and perpendicular components relative to a surface or direction of movement.

How is vector projection used in computer graphics?

In computer graphics, vector projection is used to calculate shadows, lighting effects, and perspectives by projecting 3D objects onto a 2D screen or surface.

Is the projection of a vector onto itself the same vector?

Yes, the projection of a vector onto itself is the vector itself because the dot product of the vector with itself gives its full magnitude.

9. Real-Life Applications of Vector Projection:

Vector projection is widely used in real-world applications, including:

1. Physics: In mechanics, vector projection helps in resolving forces into components that act in different directions, such as in inclined planes.

2. Computer Graphics: In 3D rendering, vector projection helps in calculating shadows and lighting effects by projecting vectors of light onto surfaces.

3. Navigation: In aviation and maritime navigation, vector projection assists in determining the velocity components of an aircraft or ship relative to wind or water currents.

4. Machine Learning: Vector projection is applied in algorithms like Principal Component Analysis (PCA) to reduce dimensionality by projecting data onto fewer axes.

10. Conclusion:

The vector projection is an essential tool in linear algebra that simplifies the relationship between vectors by projecting one vector onto another. It has far-reaching applications in physics, engineering, computer science, and data analysis. By mastering the concept of vector projection, you can solve a wide range of problems involving vector components, making it an invaluable skill in both academic and practical fields.

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