Finding Resultant Vector A 6.0 Km NE, B 4.5 Km, C 4.0 Km Physics Guide
In the realm of physics, understanding vectors is paramount, especially when dealing with motion, forces, and fields. A vector, unlike a scalar, possesses both magnitude and direction, making it crucial for accurately describing physical quantities. When multiple vectors act on an object, the combined effect is represented by the resultant vector. This article delves into the process of finding the resultant vector, specifically addressing the scenario presented where three vectors – A, B, and C – are given with their respective magnitudes and directions. By the end of this comprehensive guide, you'll have a solid understanding of how to calculate the resultant vector using component resolution and vector addition, enhancing your problem-solving skills in physics.
Before we dive into the specifics of finding the resultant vector for the given problem, it's essential to grasp the fundamental concepts and methods involved. Vector addition is not as straightforward as scalar addition due to the directional component. We can't simply add the magnitudes of the vectors; instead, we need to consider their directions as well. The most common and effective method for vector addition involves resolving vectors into their components along orthogonal axes, typically the x and y axes. This allows us to treat each direction independently, making the addition process much simpler and more accurate. Once we've found the components of the resultant vector, we can use the Pythagorean theorem and trigonometric functions to determine its magnitude and direction, providing a complete solution to the problem.
The problem presents us with three vectors: A, B, and C, each with its own magnitude and direction. Vector A has a magnitude of 6.0 km and points Northeast (NE). Vector B has a magnitude of 4.5 km and is directed 33° south of east. Vector C has a magnitude of 4.0 km and is directed 17° north of east. Our goal is to find the resultant vector, which means we need to determine both its magnitude and direction. To achieve this, we'll use the component method, which involves breaking each vector into its x and y components, adding the components separately, and then reconstructing the resultant vector from these summed components. This approach simplifies the problem and allows us to handle the directional aspects effectively.
The first step in finding the resultant vector is to resolve each vector into its x and y components. This involves using trigonometric functions, sine and cosine, to find the projections of each vector along the x and y axes. For vector A, which points Northeast, we can assume it makes a 45° angle with both the x and y axes. Therefore, its x-component (Ax) and y-component (Ay) can be calculated using the formulas: Ax = A * cos(45°) and Ay = A * sin(45°). Similarly, for vector B, which is 33° south of east, we can find its x-component (Bx) using B * cos(33°) and its y-component (By) using -B * sin(33°). The negative sign for By indicates that it points in the negative y-direction. For vector C, which is 17° north of east, we find its x-component (Cx) using C * cos(17°) and its y-component (Cy) using C * sin(17°). By calculating these components, we break down each vector into manageable parts, setting the stage for vector addition.
Now that we understand the concept of resolving vectors into components, let's perform the actual calculations for vectors A, B, and C. This step is crucial for accurately determining the resultant vector. We'll use the trigonometric functions sine and cosine, along with the given magnitudes and directions, to find the x and y components of each vector. This process will allow us to treat each direction independently, making the vector addition process much simpler and more precise. Remember, the accuracy of these calculations directly impacts the final result, so it's important to pay close attention to detail.
For vector A, with a magnitude of 6.0 km pointing Northeast, we assume a 45° angle with both the x and y axes. The x-component (Ax) is calculated as Ax = 6.0 km * cos(45°) ≈ 4.24 km. Similarly, the y-component (Ay) is calculated as Ay = 6.0 km * sin(45°) ≈ 4.24 km. Notice that both components are positive, as the vector points in both the positive x and positive y directions. For vector B, with a magnitude of 4.5 km directed 33° south of east, the x-component (Bx) is calculated as Bx = 4.5 km * cos(33°) ≈ 3.77 km. The y-component (By) is calculated as By = -4.5 km * sin(33°) ≈ -2.45 km. The negative sign indicates that the y-component points in the negative y-direction. For vector C, with a magnitude of 4.0 km directed 17° north of east, the x-component (Cx) is calculated as Cx = 4.0 km * cos(17°) ≈ 3.83 km. The y-component (Cy) is calculated as Cy = 4.0 km * sin(17°) ≈ 1.17 km. These calculations provide us with the necessary components to proceed with vector addition.
With the x and y components of each vector calculated, the next step is to add these components to find the x and y components of the resultant vector. This is a straightforward process of summing the x-components together and the y-components together. This method simplifies the vector addition by treating each direction independently, allowing us to accurately determine the overall effect of the three vectors. The resultant components will then be used to find the magnitude and direction of the resultant vector, giving us the final answer.
The x-component of the resultant vector (Rx) is found by adding the x-components of vectors A, B, and C: Rx = Ax + Bx + Cx ≈ 4.24 km + 3.77 km + 3.83 km ≈ 11.84 km. Similarly, the y-component of the resultant vector (Ry) is found by adding the y-components of vectors A, B, and C: Ry = Ay + By + Cy ≈ 4.24 km + (-2.45 km) + 1.17 km ≈ 2.96 km. These values, Rx ≈ 11.84 km and Ry ≈ 2.96 km, represent the x and y components of the resultant vector. They tell us the net displacement in the x and y directions, which we can then use to find the magnitude and direction of the resultant vector. This step is crucial in understanding the overall effect of the three vectors acting together.
Now that we have the x and y components of the resultant vector, the final step is to determine its magnitude and direction. The magnitude tells us the overall length or size of the resultant vector, while the direction tells us the angle it makes with the positive x-axis. These two values completely define the resultant vector, giving us a comprehensive understanding of its effect. We'll use the Pythagorean theorem to find the magnitude and the arctangent function to find the direction, completing our solution to the problem.
The magnitude of the resultant vector (R) can be calculated using the Pythagorean theorem: R = √(Rx² + Ry²) ≈ √((11.84 km)² + (2.96 km)²) ≈ √(140.18 km² + 8.76 km²) ≈ √148.94 km² ≈ 12.20 km. This value represents the overall displacement caused by the combined effect of the three vectors. The direction of the resultant vector (θ) can be calculated using the arctangent function: θ = arctan(Ry / Rx) ≈ arctan(2.96 km / 11.84 km) ≈ arctan(0.25) ≈ 14.04°. This angle is measured counterclockwise from the positive x-axis. Therefore, the resultant vector has a magnitude of approximately 12.20 km and a direction of approximately 14.04° north of east. This completes our analysis, providing us with a full understanding of the resultant vector's magnitude and direction.
In conclusion, we have successfully found the resultant vector for the given problem by breaking down each vector into its x and y components, adding the components separately, and then using the Pythagorean theorem and arctangent function to determine the magnitude and direction of the resultant. This process highlights the importance of understanding vector addition and component resolution in physics. The resultant vector, with a magnitude of approximately 12.20 km and a direction of 14.04° north of east, represents the combined effect of the three individual vectors, providing a complete picture of their overall impact. This method can be applied to a wide range of physics problems involving vector quantities, making it a fundamental skill for any physics student or professional.
The significance of finding the resultant vector lies in its ability to simplify complex scenarios involving multiple vectors. Instead of dealing with several vectors acting at different angles, we can represent their combined effect with a single vector. This is particularly useful in fields like mechanics, electromagnetism, and fluid dynamics, where multiple forces, fields, or velocities may be acting on an object or system. By finding the resultant vector, we can predict the overall motion or behavior of the object or system, making it a powerful tool for analysis and problem-solving. The component method, as demonstrated in this article, provides a systematic and accurate way to find the resultant vector, ensuring that both magnitude and direction are correctly accounted for. Mastering this technique is crucial for anyone seeking a deeper understanding of physics and its applications.
Original Keywords: Find the resultant vector. A = 6.0 km NE B = 4.5 km 33° south of east C = 4.0 km 17° north of east.
Repaired Keywords: How to find the resultant vector given A = 6.0 km NE, B = 4.5 km 33° south of east, and C = 4.0 km 17° north of east?
