Solving for M When Two Vectors Are Perpendicular
Understanding the concept of perpendicular vectors is fundamental in vector algebra. When two vectors are perpendicular to each other, their dot product is zero. This article will guide you through the process of determining the value of m when the vectors 3mathbf{i} - 2mathbf{j} and 2mathbf{i} - mmathbf{j} are perpendicular.
The Dot Product Method
To find the value of m, we start by using the condition for perpendicularity, which states that the dot product of the vectors must be zero. The dot product of two vectors mathbf{A} 3mathbf{i} - 2mathbf{j} and mathbf{B} 2mathbf{i} - mmathbf{j} is calculated as follows:
[mathbf{A} cdot mathbf{B} (3)(2) (-2)(-m)]
This simplifies to:
[6 2m 0]
Solving for m involves isolating it on one side of the equation:
[2m -6]
[m -3]
Hence, the value of m that makes the vectors perpendicular is -3.
Alternative Methods for Determining M
There are several ways to approach this problem, and the method can vary based on the information provided. Here are some alternative methods to solve for m:
Slope Method: One common method involves using the slopes of the vectors. The slope of the vector 3mathbf{i} - 2mathbf{j} is -frac{2}{3}. For two vectors to be perpendicular, their slopes must be negative reciprocals of each other. Therefore, the slope of 2mathbf{i} - mmathbf{j} must be frac{3}{2}. From this, we can deduce that:
[2 -m]
[m -3]
Vector Equation Method: We can also use the vector equation to find m. If the vectors are perpendicular, their dot product must be zero:
[3mathbf{i} - 2mathbf{j} cdot 2mathbf{i} - mmathbf{j} 0]
This results in:
[3(2) (-2)(-m) 0]
[6 2m 0]
From this, we again find:
[m -3]
Component Form Method: Another method involves directly using the components of the vectors. If the dot product of the vectors is zero, then:
[3(2) (-2)(-m) 0]
[6 2m 0]
[2m -6]
[m -3]
These methods all lead to the same conclusion that the value of m is -3.
Conclusion
Understanding how to determine the value of m when two vectors are perpendicular is crucial in vector algebra. Whether through the dot product, slope, or component form, the process remains consistent and straightforward. This knowledge can be applied in various fields, including physics and engineering, where vector operations play a significant role.