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Learn about the Vector Projection Formula and know how to find the projection of one vector onto another effortlessly. Our comprehensive guide simplifies the process, making it easy to understand and apply in various fields.
Contents
Vector Projection Formula
The vector projection is a mathematical operation that involves finding the component of one vector onto another vector. It is used to determine how much of one vector lies in the direction of another vector. The vector projection of vector A onto vector B is denoted as proj<sub>B</sub>A.
If A and B are two non-zero vectors, the formula for the vector projection proj<sub>B</sub>A is:
- proj<sub>B</sub>A = ((A · B) / (B · B)) B
Where:
A · B represents the dot product of vectors A and B.
B · B represents the dot product of vector B with itself (magnitude of B squared).
The result of the vector projection, proj<sub>B</sub>A, is a vector that points in the same direction as B and represents the component of A in the direction of B.
Note that if A and B are orthogonal (perpendicular), then their dot product A · B will be zero, and the vector projection will also be zero. This is because there is no projection of A onto B when they are orthogonal.
What is a Vector Projection?
Vector projection is a mathematical operation that involves projecting one vector onto another. The result of the projection is a new vector that represents the component of the original vector that lies in the direction of the target vector.
Let’s say you have two vectors: vector A and vector B. The projection of vector A onto vector B, denoted as proj(A, B), is a new vector C, which is parallel to vector B and represents how much of vector A lies in the direction of vector B.
The formula for calculating the vector projection of A onto B is as follows:
- proj(A, B) = (dot(A, B) / dot(B, B)) * B
Where:
dot(A, B) represents the dot product of vectors A and B.
dot(B, B) represents the dot product of vector B with itself (magnitude squared of vector B).
To break down the formula:
- Calculate the dot product of A and B.
- Calculate the dot product of B with itself to get its magnitude squared.
- Divide the dot product of A and B by the magnitude squared of B.
- Multiply the result by vector B to get the projected vector.
The vector projection can be used in various applications, such as physics, computer graphics, and linear algebra, where understanding the component of a vector in a particular direction is necessary.
What is the Formula for Projection of Vectors?
The projection of one vector onto another is a fundamental concept in linear algebra. Given two vectors, let’s call them vector A (the vector to be projected) and vector B (the target vector onto which the projection is to be made), the formula for projecting vector A onto vector B is:
- Projection of A onto B = ((A ⋅ B) / ||B||^2) × B
Where:
A ⋅ B denotes the dot product of vector A and vector B.
||B||^2 represents the squared magnitude (length) of vector B.
The dot product of two vectors A and B, denoted as A ⋅ B, is a scalar quantity equal to the product of the magnitudes of the vectors and the cosine of the angle between them.
Alternatively, you can express the formula using the unit vector of B, denoted as u_B:
- Projection of A onto B = (A ⋅ u_B) × u_B
Here, u_B is the unit vector of B, which is obtained by dividing B by its magnitude:
In this form, you first find the unit vector of B, and then project vector A onto it by taking the dot product of A and the unit vector of B, and finally multiplying the result by the unit vector of B.
The projection of vector A onto vector B gives a new vector that lies in the direction of vector B and represents the component of vector A in that direction.
Derivation of Vector Projection Formula
To derive the formula for vector projection, let’s consider two vectors: a vector A and a vector B. The projection of vector A onto vector B is a vector that represents the component of A that lies in the direction of B.
The vector projection of A onto B is given by the formula:
- Proj<sub>B</sub>A = ((A · B) / (B · B)) B
where:
Proj<sub>B</sub>A is the vector projection of A onto B.
A · B represents the dot product of vectors A and B.
B · B represents the dot product of vector B with itself (magnitude squared of vector B).
Now, let’s derive this formula step by step:
Step 1: Find the scalar projection of A onto B
The scalar projection of A onto B is the length of the projection of A onto B. It is given by the formula:
- Scalar_proj<sub>B</sub>A = (A · B) / |B|
where |B| represents the magnitude of vector B.
Step 2: Find the direction vector of the projection
The direction vector of the projection, Proj<sub>B</sub>A, is the unit vector in the direction of B. It is given by:
- Proj<sub>B</sub>A = (1 / |B|) B
Step 3: Combine the scalar projection and direction vector
Finally, the vector projection of A onto B is obtained by multiplying the scalar projection with the direction vector:
Proj<sub>B</sub>A = Scalar_proj<sub>B</sub>A × Proj<sub>B</sub>A
= ((A · B) / |B|) × (1 / |B|) B
= ((A · B) / (B · B)) B
Thus, the formula for the vector projection of A onto B is:
- Proj<sub>B</sub>A = ((A · B) / (B · B)) B
This formula gives us the vector that represents the projection of A onto B. The projection vector has the same direction as B and its magnitude is proportional to the dot product of A and B, relative to the magnitude of B squared.
What is the Vector Projection of A onto B?
The vector projection of vector A onto vector B, denoted as proj(A, B), is a component of vector A that lies in the direction of vector B. In other words, it’s the vector that represents the “shadow” of vector A cast onto vector B.
The formula to calculate the vector projection of A onto B is:
- proj(A, B) = (dot product(A, B) / ||B||^2) * B
where:
dot product(A, B) is the dot product between vectors A and B.
||B||^2 is the squared magnitude of vector B.
denotes scalar multiplication.
Here’s a step-by-step explanation of the process:
- Calculate the dot product between A and B: dot product(A, B) = A · B
- Calculate the squared magnitude of B: ||B||^2 = (B · B)
- Divide the dot product by the squared magnitude of B: dot product(A, B) / ||B||^2
- Multiply the result by vector B to get the projection: (dot product(A, B) / ||B||^2) * B
- The resulting vector proj(A, B) will be the vector projection of A onto B. It represents the part of A that points in the direction of B.
Solved Problems on Vector Projection
Let’s go through a few solved problems on vector projection. Vector projection involves finding the component of one vector onto another. The projection of vector A onto vector B is denoted as projB(A). The formula for vector projection is:
- projB(A) = (A dot B / |B|^2) * B
Where:
- A is the vector you want to project.
- B is the vector onto which you want to project A.
- A dot B is the dot product of A and B.
- |B|^2 is the squared magnitude (length) of B.
Let’s dive into some examples:
Problem 1:
Find the projection of vector A = (3, 2) onto vector B = (1, 4).
Solution:
Step 1: Calculate the dot product of A and B.
A dot B = (3 * 1) + (2 * 4) = 3 + 8 = 11
Step 2: Calculate the squared magnitude of B.
|B|^2 = √(1^2 + 4^2) = √(1 + 16) = √17
Step 3: Calculate the projection of A onto B.
projB(A) = (A dot B / |B|^2) * B = (11 / √17) * (1, 4)
Approximating the value of (11 / √17) ≈ 2.67, the projection of A onto B is approximately (2.67, 10.68).
Problem 2:
Find the projection of vector C = (6, -3, 2) onto vector D = (-2, 1, 2).
Solution:
Step 1: Calculate the dot product of C and D.
C dot D = (6 * -2) + (-3 * 1) + (2 * 2) = -12 – 3 + 4 = -11
Step 2: Calculate the squared magnitude of D.
|D|^2 = √((-2)^2 + 1^2 + 2^2) = √(4 + 1 + 4) = √9 = 3
Step 3: Calculate the projection of C onto D.
projD(C) = (C dot D / |D|^2) * D = (-11 / 3) * (-2, 1, 2)
Multiplying the scalar -11/3 with each component of D, we get the projection as (22/3, -11/3, -22/3).
These are two examples of how to find vector projections. Always remember to compute the dot product and magnitude accurately for correct results.
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