Vector Addition and Magnitude Analysis: Exploring the Resultant of Vectors A and B

In this article, we will delve into a specific vector addition problem involving two vectors, A and B. We will explore the magnitude, direction, and resultant of these vectors, providing a comprehensive understanding of vector operations and their applications in physics and mathematics.

Understanding Vectors A and B

We are given two vectors: Vector A A -14j and Vector B B 28i. Vector A points in the negative y-direction and has a magnitude of 14 units, while Vector B has a magnitude twice that of Vector A and points in the positive x-direction.

Vector Addition and the Resultant

When adding vectors, we consider their components. Vector addition is a linear operation that combines the components of the vectors to obtain the resultant vector. In this case, we will add Vector A and Vector B to find the resultant vector, denoted as Vector C.

First, let's express both vectors in their component forms: Vector A: A 0i - 14j (x-component: 0, y-component: -14) Vector B: B 28i 0j (x-component: 28, y-component: 0) Now, we add the components of the vectors. Resultant in x-direction (i-component): 0 28 28 Resultant in y-direction (j-component): -14 0 -14 Therefore, the resultant Vector C can be written as: C 28i - 14j

Magnitude of the Resultant Vector C

The magnitude of the resultant vector is calculated using the Pythagorean theorem, as the components of Vector C form a right triangle with the resultant vector as the hypotenuse. The magnitude represents the total length of the vector:

begin{align*}text{Magnitude of C} sqrt{(28)^2 (-14)^2} sqrt{784 196} sqrt{980} approx 31.30end{align*}

Direction of the Resultant Vector C

The direction of the resultant vector C (counterclockwise from the positive x-axis) can be found using the inverse tangent function:

theta arctanleft(frac{text{y-component}}{text{x-component}}right) arctanleft(frac{-14}{28}right) arctanleft(-frac{1}{2}right)

Since the resultant vector C is in the fourth quadrant (positive x, negative y), we need to adjust the angle to be counterclockwise from the positive x-axis:

(theta 360^circ arctan(-frac{1}{2}) approx 360^circ - 26.57^circ 333.43^circ)

Conclusion

In conclusion, the resultant vector C of Vectors A and B has a magnitude of approximately 31.30 units and points in a direction of about 333.43 degrees counterclockwise from the positive x-axis. Understanding vector addition and the resultant vector is fundamental in various fields, including physics, engineering, and computer graphics. By mastering these concepts, you can solve complex problems involving vector operations effectively.

Feel free to explore more examples and exercises on vector addition, magnitude, and direction to further enhance your understanding. Vector operations are a crucial tool in many scientific and technical applications.