Exploring Vectors Perpendicular to a Given Vector A

Introduction

Understanding the concept of vectors perpendicular to a given vector is fundamental in vector algebra and has applications in various fields, including physics and engineering. In this article, we will explore the vectors that are perpendicular to the given vector

Concept of Perpendicular Vectors

To determine if two vectors are perpendicular, we use the dot product. Two vectors and are perpendicular if and only if their dot product is zero. In mathematical terms, if , then and are perpendicular.

Deriving Perpendicular Vectors

Let's consider the vector . We want to find vectors that are perpendicular to .

Step-by-Step Derivation

The dot product of and is given by:

Setting the dot product to zero to find perpendicular vectors:

4x - 3y 0z 0 or 4x - 3y 0

Rewriting the equation:

3y 4x or y -frac{4}{3}x

This equation implies that any vector of the form is perpendicular to .

Specific Instances

Choosing two specific values for x:

x 3, then y -4. Vector x 1, then y -frac{4}{3}. Vector

Thus, two vectors perpendicular to are:

Understanding Infinite Perpendicular Vectors

There are not just two vectors but an infinite number of perpendicular vectors to . This is because any plane that is perpendicular to the direction of will contain vectors that are perpendicular to .

Intuitive Explanation

Imagine vector as a line in space. Any plane perpendicular to this line will contain vectors that are perpendicular to . Let's denote this plane as P, which passes through the origin (0,0,0).

Any point on the plane P can be represented as (x, y, z). The vectors originating from the origin and ending at these points are of the form xmathbf{i} ymathbf{j} zmathbf{k} . For these vectors to be perpendicular to , their dot product with must be zero:

(4mathbf{i} - 3mathbf{j} 0mathbf{k}) cdot (xmathbf{i} ymathbf{j} zmathbf{k}) 4x - 3y 0z 0

This simplifies to:

4x - 3y 0

This equation implies that for any value of z, x and y must satisfy the relationship 4x 3y.

Examples of Infinite Perpendicular Vectors

Some examples of vectors that are perpendicular to are:

-3mathbf{i} 4mathbf{j} rmathbf{k} where r is any real number. -3mathbf{i} 4mathbf{j} 1mathbf{k} -3mathbf{i} 4mathbf{j} 5mathbf{k}

These vectors are all of the form -3rmathbf{i} 4rmathbf{j} rmathbf{k} for any real number r.

Conclusion

In summary, to find vectors that are perpendicular to a given vector , we use the concept of the dot product and solve the equation derived from it. This results in an infinite set of vectors that are perpendicular to .

Understanding this concept is crucial for various applications in mathematics, physics, and engineering. By exploring the vectors perpendicular to a given vector, we gain insight into the geometric and algebraic properties of vectors in space.