Determining a vector that is perpendicular to another vector is a fundamental concept in linear algebra and has numerous applications in various fields, including physics, engineering, and computer graphics. The process involves understanding the properties of dot products and the geometrical relationship between orthogonal vectors. This article will delve into the method of finding a unit vector perpendicular to a given vector, exploring the underlying principles and providing a step-by-step guide for accomplishing this task.
Understanding Perpendicular Vectors
Two vectors are considered perpendicular, or orthogonal, if their dot product is equal to zero. The dot product, often denoted by a dot (·), is a mathematical operation that takes two vectors and produces a scalar value. This value is determined by multiplying the corresponding components of the two vectors and summing the results. Mathematically, if we have two vectors u = (u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>) and v = (v<sub>1</sub>, v<sub>2</sub>, v<sub>3</sub>), their dot product is:
u · v = u<sub>1</sub>v<sub>1</sub> + u<sub>2</sub>v<sub>2</sub> + u<sub>3</sub>v<sub>3</sub>
For u and v to be perpendicular, this dot product must equal zero. This principle serves as the foundation for finding a vector perpendicular to a given vector.
The Process of Finding a Perpendicular Vector
To determine a vector perpendicular to a given vector a, we can follow these steps:
- Choose a candidate vector: Begin by selecting any vector b that is not a scalar multiple of a. This ensures that b is not parallel to a.
- Calculate the dot product: Find the dot product of a and b.
- Adjust b to make it orthogonal: If the dot product from step 2 is not zero, we need to modify b to ensure it becomes perpendicular to a. This involves scaling b by a factor that makes the dot product equal to zero. We can achieve this by subtracting the projection of b onto a from b.
Projection of One Vector onto Another
The projection of b onto a is the component of b that lies in the direction of a. It can be calculated using the formula:
proj<sub>a</sub>b = (a · b / ||a||<sup>2</sup>) * a
where:
- proj<sub>a</sub>b is the projection of b onto a
- a · b is the dot product of a and b
- ||a|| is the magnitude of a
Adjusting b to Be Perpendicular
Now, to find the vector c that is perpendicular to a, we subtract the projection of b onto a from b:
c = b - proj<sub>a</sub>b
This vector c will be orthogonal to a, as its dot product with a will be zero.
Finding a Unit Vector Perpendicular to Another Vector
A unit vector is a vector with a magnitude of 1. To find a unit vector u perpendicular to a, we follow the steps described above to find a vector c perpendicular to a and then normalize it by dividing by its magnitude.
u = c / ||c||
Example: Finding a Unit Vector Perpendicular to (1, 2, 3)
Let's consider the vector a = (1, 2, 3) and find a unit vector u that is perpendicular to it.
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Choose a candidate vector: We select b = (0, 1, 0). This vector is not a scalar multiple of a.
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Calculate the dot product: a · b = (1)(0) + (2)(1) + (3)(0) = 2.
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Adjust b to make it orthogonal: We need to find the projection of b onto a:
proj<sub>a</sub>b = (a · b / ||a||<sup>2</sup>) * a = (2 / (1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup>)) * (1, 2, 3) = (1/7, 2/7, 3/7)
Now, we subtract the projection from b:
c = b - proj<sub>a</sub>b = (0, 1, 0) - (1/7, 2/7, 3/7) = (-1/7, 5/7, -3/7)
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Normalize c to find a unit vector:
||c|| = √((-1/7)<sup>2</sup> + (5/7)<sup>2</sup> + (-3/7)<sup>2</sup>) = √(35/49) = √35 / 7
u = c / ||c|| = (-1/7, 5/7, -3/7) / (√35 / 7) = (-1/√35, 5/√35, -3/√35)
Therefore, a unit vector perpendicular to (1, 2, 3) is u = (-1/√35, 5/√35, -3/√35).
Conclusion
Finding a unit vector perpendicular to another vector is a fundamental concept in linear algebra and has significant practical implications in various fields. The process involves utilizing the dot product property of orthogonal vectors and involves calculating the projection of one vector onto another. By following the step-by-step guide outlined in this article, you can effectively determine a unit vector perpendicular to any given vector. This knowledge empowers you to analyze and manipulate vectors in diverse applications, from physics and engineering to computer graphics and beyond.
