Orthogonal Vectors: Finding K Value
Let's dive into the world of vectors and orthogonality! In this article, we'll explore how to determine the value of a real number, denoted as k, that makes two vectors orthogonal. Specifically, we'll focus on vectors A = (2, k, 3) and B = (1, 4, k). By the end of this guide, you'll understand how to use the concept of the dot product (or scalar product) to find the possible values of k.
Understanding Orthogonality and the Dot Product
So, what does it mean for two vectors to be orthogonal? Simply put, orthogonal vectors are perpendicular to each other. In other words, the angle between them is 90 degrees. A crucial property that helps us identify orthogonality is the dot product.
The dot product of two vectors, say A and B, is defined as:
A · B = |A| |B| cos(θ)
Where |A| and |B| represent the magnitudes (lengths) of vectors A and B, respectively, and θ is the angle between them.
When vectors A and B are orthogonal, the angle θ is 90 degrees, and cos(90°) = 0. Therefore, the dot product of two orthogonal vectors is always zero.
A · B = 0 (if A and B are orthogonal)
This property provides us with a straightforward method to check if two vectors are orthogonal: calculate their dot product. If the result is zero, the vectors are orthogonal. If not, they are not orthogonal.
For our specific problem, we are given vectors A = (2, k, 3) and B = (1, 4, k). To find the value(s) of k that make these vectors orthogonal, we need to calculate their dot product and set it equal to zero.
The dot product of A and B is calculated as follows:
A · B = (2)(1) + (k)(4) + (3)(k)
A · B = 2 + 4k + 3k
Combining like terms, we get:
A · B = 2 + 7k
Now, for A and B to be orthogonal, their dot product must be zero. So, we set the expression equal to zero and solve for k:
2 + 7k = 0
7k = -2
k = -2/7
Therefore, the value of k that makes vectors A and B orthogonal is -2/7. This result means that if k is equal to -2/7, the vectors A = (2, -2/7, 3) and B = (1, 4, -2/7) are perpendicular to each other. This is the only value of k that satisfies the condition of orthogonality for these two vectors. If k is any other value, the dot product of A and B will not be zero, indicating that the vectors are not orthogonal.
Calculating the Dot Product
To determine if the vectors A = (2, k, 3) and B = (1, 4, k) are orthogonal, we need to calculate their dot product. The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It provides a way to determine the angle between two vectors, and most importantly for our problem, it tells us whether the vectors are orthogonal (perpendicular).
The dot product of two vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) is defined as:
A · B = a₁b₁ + a₂b₂ + a₃b₃
In our case, A = (2, k, 3) and B = (1, 4, k). Plugging these values into the formula, we get:
A · B = (2)(1) + (k)(4) + (3)(k)
Let's break this down step by step:
- Multiply the first components of the vectors: (2)(1) = 2
- Multiply the second components of the vectors: (k)(4) = 4k
- Multiply the third components of the vectors: (3)(k) = 3k
Now, add these products together:
A · B = 2 + 4k + 3k
Combine the terms with k:
A · B = 2 + 7k
This expression, 2 + 7k, represents the dot product of vectors A and B. For the vectors to be orthogonal, this dot product must equal zero.
So, we set the dot product equal to zero:
2 + 7k = 0
Now, we solve for k:
7k = -2
k = -2/7
Thus, the value of k that makes the vectors A and B orthogonal is -2/7. This means that if we substitute -2/7 for k in the vectors A and B, the angle between them will be 90 degrees, indicating that they are perpendicular. If k has any other value, the vectors will not be orthogonal.
Solving for k
To find the value of k that makes vectors A = (2, k, 3) and B = (1, 4, k) orthogonal, we must set their dot product equal to zero and solve the resulting equation. As we've already established, the dot product of A and B is given by:
A · B = (2)(1) + (k)(4) + (3)(k) = 2 + 4k + 3k = 2 + 7k
For A and B to be orthogonal, their dot product must be zero. Therefore, we set the expression equal to zero:
2 + 7k = 0
Now, we solve for k:
- Subtract 2 from both sides of the equation:
7k = -2
- Divide both sides by 7 to isolate k:
k = -2/7
Therefore, the value of k that makes the vectors A = (2, k, 3) and B = (1, 4, k) orthogonal is k = -2/7. This means that when k is equal to -2/7, the angle between vectors A and B is 90 degrees, and they are perpendicular.
To verify this result, we can substitute k = -2/7 back into the vectors A and B and calculate their dot product:
A = (2, -2/7, 3)
B = (1, 4, -2/7)
A · B = (2)(1) + (-2/7)(4) + (3)(-2/7)
A · B = 2 - 8/7 - 6/7
A · B = 2 - 14/7
A · B = 2 - 2
A · B = 0
Since the dot product is indeed zero, our value of k = -2/7 is correct.
It's important to note that orthogonality is a specific condition. If k were any value other than -2/7, the dot product of A and B would not be zero, indicating that the vectors are not orthogonal. The value k = -2/7 is the only solution that satisfies the orthogonality condition for the given vectors A and B.
Conclusion
In summary, we determined the value of k that makes the vectors A = (2, k, 3) and B = (1, 4, k) orthogonal by using the concept of the dot product. We found that the dot product of two orthogonal vectors is zero, which allowed us to set up and solve an equation for k. The solution we obtained was k = -2/7. This is the only value of k that ensures the vectors A and B are perpendicular to each other. Understanding the relationship between the dot product and orthogonality is fundamental in vector algebra and has numerous applications in various fields, including physics, engineering, and computer graphics.