Determining k for Orthogonal Vectors: A Detailed Guide

In this article, we will explore how to determine the value of k such that two vectors u and v are orthogonal. Specifically, we will work through the example given and explain the required steps in detail. This problem involves vector operations and the dot product, which are fundamental concepts in linear algebra.

Understanding the Vectors

Let's consider the vectors u and v given as:

u -5k i 4j

v 21i - 3j

At first glance, you might confuse these with quaternions. However, in this context, these are vectors in two-dimensional space. The notation may seem unfamiliar, but the underlying concepts are based on vector operations.

The Condition for Orthogonality

Two vectors are orthogonal if their dot product is zero. Mathematically, for vectors u and v, this condition is expressed as:

u · v 0

Let's compute the dot product of u and v step by step.

Computing the Dot Product

The dot product of two vectors u uxi uyj and v vxi vyj is given by:

u · v uxvx uyvy

In our case:

ux -5k uy 4 vx 21 vy -3

Substituting these values into the dot product formula, we get:

u · v (-5k)(21) (4)(-3)

Simplifying this expression:

u · v -105k - 12

Solving for k

For u and v to be orthogonal, their dot product must equal zero:

-105k - 12 0

Solving for k involves isolating k in the equation:

-105k 12

k -12 / 105

Simplifying the fraction:

k -4 / 35

Conclusion

The value of k that makes the vectors u -5k i 4j and v 21i - 3j orthogonal is -4/35. This solution demonstrates the application of the dot product to determine orthogonality in vector spaces.

Further Reading

To gain a deeper understanding of vector operations and the dot product, consider exploring the following topics:

Orthogonality in Vector Spaces Dot Product and Its Applications Vector Algebra: A Comprehensive Guide

Understanding these concepts will not only help in solving similar problems but also provide a solid foundation for advanced topics in linear algebra and multivariable calculus.

Key Takeaways

Vectors can be orthogonal if their dot product is zero. The dot product of two vectors u uxi uyj and v vxi vyj is given by u · v uxvx uyvy. Solving for k in the equation -105k - 12 0 yields k -4/35 for the vectors to be orthogonal.

By following these steps and understanding the principles involved, you can confidently solve similar problems involving orthogonal vectors.