How to Find a Vector Perpendicular to Another Vector

Have you ever wondered how to determine a vector that is perpendicular to another vector? This guide will walk you through the process in detail, focusing on both 2D and 3D spaces. Whether you're dealing with basic vector operations or more advanced concepts like the cross product, this article will provide you with comprehensive insights and practical examples.

Understanding Vector Perpendicularity

Two vectors are said to be perpendicular if the angle between them is 90 degrees. In mathematical terms, this can be expressed as the dot product of two perpendicular vectors being zero. The dot product of vectors u and v in n-dimensional space is given by:

u · v |u| |v| cos(θ)

Where θ is the angle between the vectors. For perpendicular vectors, θ 90°, hence cos(90°) 0, making the dot product u · v 0.

2D Perpendicular Vectors

In a two-dimensional (2D) space, finding a vector perpendicular to another is a straightforward process. Given a vector v (x, y), you can obtain a vector u perpendicular to v by swapping the components and changing the sign of one of them. The resulting vector u will be:

Example:

Given v (3, 4), a perpendicular vector will be u (-4, 3).

3D Perpendicular Vectors Using Cross Product

In a three-dimensional (3D) space, the process becomes a bit more complex but still manageable. To find a vector perpendicular to another vector v (x, y, z), you can use the cross product with a non-parallel vector. If v is not aligned with the x-axis, you can use the vector w (1, 0, 0) and find the perpendicular vector u as:

u v × w

The cross product of vectors v (x, y, z) and w (1, 0, 0) is calculated as:

u (0, -z, y)

Example:

Given v (1, 2, 3) and w (1, 0, 0), the perpendicular vector u can be found as:

u (1, 2, 3) × (1, 0, 0) (0, -3, 2)

General Orthogonal Vectors

In most contexts, there are many vectors that are orthogonal to any specific vector you choose. However, there is a simple and foolproof way to find a vector orthogonal to a given vector v in any vector space equipped with a notion of orthogonality. Just taking the zero vector will suffice since it is orthogonal to every vector, including itself. The zero vector 0 (0, 0, 0) satisfies the condition for orthogonality with any vector v.

Conclusion

Understanding how to find a vector perpendicular to another vector is crucial in many fields, from physics to computer science. Whether you're working in 2D space with simple swaps or in 3D space with cross products, this guide provides a comprehensive overview of the methods and techniques involved. By familiarizing yourself with these concepts, you'll be well-equipped to handle a wide range of vector operations.

Related Keywords: vector perpendicular, cross product, Gram-Schmidt process