Vector Operations And Simplification Finding BA And AC

In coordinate geometry, vectors play a crucial role in representing magnitude and direction. They allow us to perform various geometric operations algebraically, making complex problems more manageable. This article delves into the fundamental concepts of vector operations, specifically focusing on vector addition, subtraction, and scalar multiplication. We will explore how to express vectors in terms of their components and how to use these components to simplify vector expressions. Understanding these concepts is essential for solving a wide range of problems in physics, engineering, and computer graphics.

The main objective of this article is to demonstrate how to find the vectors $\overrightarrow{BA}$ and $\overrightarrow{AC}$ given the vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{CO}$. This involves using vector subtraction and the properties of vector addition. We will break down the process step by step, providing a clear and concise explanation of each step. By the end of this article, readers will have a solid understanding of how to perform vector operations and simplify vector expressions effectively. We will also emphasize the importance of visualizing vectors and their operations in a coordinate plane to enhance understanding and problem-solving skills.

Problem Statement

Given the position vectors $\overrightarrow{OA} = 11x + 6y$, $\overrightarrow{OB} = 4x + 10y$, and $\overrightarrow{CO} = -13x + 11y$, our task is to find the vectors $\overrightarrow{BA}$ and $\overrightarrow{AC}$ in their simplest forms. This problem exemplifies how vector operations, such as addition and subtraction, can be applied to determine the vectors connecting different points in a coordinate system. The solution involves understanding the relationships between position vectors and displacement vectors, and how to manipulate them algebraically. This type of problem is fundamental in vector algebra and has applications in various fields, including physics and engineering, where understanding the relationships between positions and displacements is crucial.

a) Finding $\overrightarrow{BA}$

To find the vector $\overrightarrow{BA}$, we need to understand the relationship between the position vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$. The vector $\overrightarrow{BA}$ represents the displacement from point B to point A. In terms of position vectors, $\overrightarrow{BA}$ can be expressed as the difference between the position vector of A and the position vector of B. Mathematically, this is written as:

$$\overrightarrow{BA} = \overrightarrow{OA} - \overrightarrow{OB}$$

This formula is a fundamental concept in vector algebra. It states that to find the vector pointing from one point to another, you subtract the position vector of the starting point from the position vector of the ending point. This concept is crucial for solving problems involving displacements and relative positions in physics and engineering. For instance, in mechanics, it is used to determine the relative velocity or displacement of objects.

Now, we substitute the given values of $\overrightarrow{OA}$ and $\overrightarrow{OB}$ into the equation:

$$\overrightarrow{BA} = (11x + 6y) - (4x + 10y)$$

To simplify this expression, we need to group the like terms. This involves combining the x-components and the y-components separately. This process is based on the distributive property of scalar multiplication over vector addition and subtraction. It is a standard technique in vector algebra and is used extensively in simplifying vector expressions.

$$\overrightarrow{BA} = 11x + 6y - 4x - 10y$$

Combining the x-terms ($11x$ and $-4x$) and the y-terms ($6y$ and $-10y$), we get:

$$\overrightarrow{BA} = (11x - 4x) + (6y - 10y)$$

Now, perform the subtraction for each component:

$$\overrightarrow{BA} = 7x - 4y$$

Thus, the vector $\overrightarrow{BA}$ in its simplest form is $7x - 4y$. This result tells us the displacement vector from point B to point A. The x-component of the vector is 7, and the y-component is -4. This means that to go from point B to point A, we need to move 7 units in the x-direction and -4 units in the y-direction. This understanding is essential for visualizing vectors and their operations in a coordinate plane.

b) Finding $\overrightarrow{AC}$

To find the vector $\overrightarrow{AC}$, we need to express it in terms of the given vectors. We know $\overrightarrow{OA}$ and $\overrightarrow{CO}$, and we need to find a relationship that connects these vectors to $\overrightarrow{AC}$. The key here is to realize that $\overrightarrow{AC}$ can be expressed as the sum of $\overrightarrow{AO}$ and $\overrightarrow{OC}$. This is based on the triangle law of vector addition, which states that if you have two vectors forming two sides of a triangle, the third side is given by the sum of those two vectors. This is a fundamental concept in vector algebra and is used extensively in physics and engineering to analyze forces and displacements.

First, we need to find $\overrightarrow{AO}$. Since $\overrightarrow{AO}$ is the reverse direction of $\overrightarrow{OA}$, it is simply the negative of $\overrightarrow{OA}$. This is a direct consequence of the definition of a vector's direction. Reversing the direction of a vector changes its sign. Mathematically, this is expressed as:

$$\overrightarrow{AO} = -\overrightarrow{OA}$$

Substituting the given value of $\overrightarrow{OA}$, we get:

$$\overrightarrow{AO} = -(11x + 6y) = -11x - 6y$$

Next, we need to find $\overrightarrow{OC}$. We are given $\overrightarrow{CO}$, which is the vector from C to O. To find $\overrightarrow{OC}$, which is the vector from O to C, we simply take the negative of $\overrightarrow{CO}$:

$$\overrightarrow{OC} = -\overrightarrow{CO}$$

Substituting the given value of $\overrightarrow{CO}$, we get:

$$\overrightarrow{OC} = -(-13x + 11y) = 13x - 11y$$

Now, we can find $\overrightarrow{AC}$ using the relationship:

$$\overrightarrow{AC} = \overrightarrow{AO} + \overrightarrow{OC}$$

This equation is a direct application of the triangle law of vector addition. It states that the vector $\overrightarrow{AC}$ is the sum of the vectors $\overrightarrow{AO}$ and $\overrightarrow{OC}$. Substituting the values we found for $\overrightarrow{AO}$ and $\overrightarrow{OC}$, we get:

$$\overrightarrow{AC} = (-11x - 6y) + (13x - 11y)$$

To simplify this expression, we group the like terms, similar to what we did in part (a):

$$\overrightarrow{AC} = -11x - 6y + 13x - 11y$$

Combining the x-terms ($-11x$ and $13x$) and the y-terms ($-6y$ and $-11y$), we get:

$$\overrightarrow{AC} = (-11x + 13x) + (-6y - 11y)$$

Now, perform the addition and subtraction for each component:

$$\overrightarrow{AC} = 2x - 17y$$

Thus, the vector $\overrightarrow{AC}$ in its simplest form is $2x - 17y$. This result tells us the displacement vector from point A to point C. The x-component of the vector is 2, and the y-component is -17. This means that to go from point A to point C, we need to move 2 units in the x-direction and -17 units in the y-direction. This understanding is crucial for visualizing vectors and their operations in a coordinate plane.

Conclusion

In conclusion, we have successfully found the vectors $\overrightarrow{BA}$ and $\overrightarrow{AC}$ in their simplest forms. For $\overrightarrow{BA}$, we used the formula $\overrightarrow{BA} = \overrightarrow{OA} - \overrightarrow{OB}$ and obtained $7x - 4y$. For $\overrightarrow{AC}$, we used the relationship $\overrightarrow{AC} = \overrightarrow{AO} + \overrightarrow{OC}$, where $\overrightarrow{AO} = -\overrightarrow{OA}$ and $\overrightarrow{OC} = -\overrightarrow{CO}$, and obtained $2x - 17y$. These results demonstrate the power of vector operations in solving geometric problems. By understanding the relationships between position vectors and displacement vectors, and by applying vector addition and subtraction, we can effectively determine the vectors connecting different points in a coordinate system.

The methods used in this article are fundamental in vector algebra and have wide-ranging applications in various fields. In physics, they are used to analyze forces, velocities, and displacements. In engineering, they are used in structural analysis, robotics, and computer graphics. The ability to manipulate vectors and simplify vector expressions is a crucial skill for anyone working in these fields. This article has provided a clear and concise explanation of these methods, equipping readers with the tools they need to tackle more complex problems involving vectors. The key takeaways from this article are the importance of understanding the triangle law of vector addition, the concept of reversing a vector's direction by changing its sign, and the technique of simplifying vector expressions by grouping like terms. These concepts and techniques are essential for mastering vector algebra and its applications.