Vector Operations Finding Resultant Vectors

Jul 11, 2025 by ADMIN 44 views

In linear algebra and physics, vectors are fundamental entities used to represent quantities that have both magnitude and direction. They are widely applied in various fields, including physics, engineering, computer graphics, and more. Vectors can be expressed in different forms, such as geometric representation and component form. When vectors are expressed in component form, we can perform various algebraic operations on them, such as addition, subtraction, and scalar multiplication. These operations allow us to combine vectors and analyze their resultant effects. This article delves into the realm of vector operations, specifically focusing on how to find the combination of vectors that yields a desired resultant vector. We'll explore the concepts of vector addition, scalar multiplication, and how to solve for unknown coefficients to achieve a specific resultant vector. Through examples and explanations, this article aims to provide a comprehensive understanding of vector operations and their applications in determining resultant vectors.

The ability to manipulate vectors and determine their resultant effects is crucial in various applications. For example, in physics, vectors are used to represent forces, velocities, and accelerations. Understanding how to add and subtract vectors allows us to analyze the net force acting on an object or the resultant velocity of a moving object. In computer graphics, vectors are used to represent the position and direction of objects in 3D space. By performing vector operations, we can manipulate these objects, rotate them, and move them around the scene. In this article, we will explore the mathematical techniques used to solve for unknown coefficients in vector equations, allowing us to express a desired resultant vector as a linear combination of given vectors.

This article addresses the following problem: Given four vectors in component form:

$\vec{a} = 4 \hat{i} - 4 \hat{j}$
$\vec{b} = -5 \hat{i} + 8 \hat{j}$
$\vec{c} = -\hat{i} - 3 \hat{j}$
$\vec{d} = 6 \hat{i} + 2 \hat{j}$

The objective is to choose the expression that produces the following resultant vector:

$\vec{r} = 6 \hat{i} + 9 \hat{j}$

This problem requires us to find a linear combination of the given vectors that results in the desired vector $\vec{r}$ . In other words, we need to find scalars (coefficients) that, when multiplied by the given vectors and added together, produce $\vec{r}$ . This problem is a fundamental example of how vector operations can be used to achieve specific outcomes, and it highlights the importance of understanding vector addition and scalar multiplication.

To effectively solve the problem at hand, a strong understanding of vector operations is essential. The two primary operations we will utilize are vector addition and scalar multiplication. These operations, when combined, allow us to manipulate vectors and express them as linear combinations, ultimately enabling us to achieve a desired resultant vector. Let's delve into each operation in detail:

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. When vectors are expressed in component form, addition is performed by adding the corresponding components together. This means that the $\hat{i}$ components are added together, and the $\hat{j}$ components are added together separately. The result is a new vector whose components are the sums of the corresponding components of the original vectors. Mathematically, if we have two vectors:

$\vec{u} = u_x \hat{i} + u_y \hat{j}$ and $\vec{v} = v_x \hat{i} + v_y \hat{j}$

Their sum, $\vec{u} + \vec{v}$ , is calculated as:

$\vec{u} + \vec{v} = (u_x + v_x) \hat{i} + (u_y + v_y) \hat{j}$

In essence, vector addition represents the combined effect of the individual vectors. Imagine two forces acting on an object; the resultant force is the vector sum of the individual forces. This concept extends beyond forces and applies to various vector quantities.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the magnitude of the vector while preserving its direction (or reversing it if the scalar is negative). To perform scalar multiplication, each component of the vector is multiplied by the scalar. If we have a vector $\vec{u} = u_x \hat{i} + u_y \hat{j}$ and a scalar $k$ , then the scalar multiplication $k\vec{u}$ is calculated as:

$k\vec{u} = (ku_x) \hat{i} + (ku_y) \hat{j}$

Scalar multiplication allows us to control the length of a vector. A scalar greater than 1 will stretch the vector, while a scalar between 0 and 1 will shrink it. A negative scalar will reverse the direction of the vector in addition to scaling its magnitude. This operation is fundamental in adjusting the contribution of a vector in a linear combination.

Linear Combinations

By combining vector addition and scalar multiplication, we can create linear combinations of vectors. A linear combination of vectors is a sum of scalar multiples of those vectors. This concept is central to solving the problem at hand. Given vectors $\vec{a}, \vec{b}, \vec{c}$ , and $\vec{d}$ , a linear combination can be expressed as:

$x\vec{a} + y\vec{b} + z\vec{c} + w\vec{d}$

where $x, y, z,$ and $w$ are scalars. Our goal is to find the specific values of these scalars that make the linear combination equal to the desired resultant vector $\vec{r}$ . This involves setting up a system of equations based on the components of the vectors and solving for the unknown scalars. The solution to this system will provide us with the coefficients that, when applied to the given vectors, produce the desired resultant vector.

Now, let's apply our understanding of vector operations to solve the problem. We are given four vectors:

$\vec{a} = 4 \hat{i} - 4 \hat{j}$
$\vec{b} = -5 \hat{i} + 8 \hat{j}$
$\vec{c} = -\hat{i} - 3 \hat{j}$
$\vec{d} = 6 \hat{i} + 2 \hat{j}$

And we want to find coefficients $x, y, z,$ and $w$ such that:

$x\vec{a} + y\vec{b} + z\vec{c} + w\vec{d} = 6 \hat{i} + 9 \hat{j}$

To solve this, we will express the left-hand side of the equation in component form and then equate the corresponding components with the right-hand side. This will give us a system of linear equations that we can solve for the unknowns.

Setting up the Equations

First, let's write out the linear combination in component form:

$x(4 \hat{i} - 4 \hat{j}) + y(-5 \hat{i} + 8 \hat{j}) + z(-\hat{i} - 3 \hat{j}) + w(6 \hat{i} + 2 \hat{j}) = 6 \hat{i} + 9 \hat{j}$

Now, distribute the scalars and group the $\hat{i}$ and $\hat{j}$ components:

$(4x - 5y - z + 6w) \hat{i} + (-4x + 8y - 3z + 2w) \hat{j} = 6 \hat{i} + 9 \hat{j}$

For the two vectors to be equal, their corresponding components must be equal. This gives us the following system of linear equations:

$4x - 5y - z + 6w = 6$ (Equation 1)
$-4x + 8y - 3z + 2w = 9$ (Equation 2)

Solving the System of Equations

We now have a system of two equations with four unknowns. This system is underdetermined, meaning there are infinitely many solutions. To find a specific solution, we can make assumptions or introduce additional constraints. For instance, we can set some of the variables to zero and solve for the others. Let's explore a few scenarios to illustrate this:

Scenario 1: Setting z = 0 and w = 0

If we set $z = 0$ and $w = 0$ , the system simplifies to:

$4x - 5y = 6$
$-4x + 8y = 9$

Adding the two equations eliminates $x$ :

$3y = 15$

So, $y = 5$ . Substituting $y = 5$ into the first equation:

$4x - 5(5) = 6$

$4x = 31$

$x = \frac{31}{4}$

Thus, one possible solution is $x = \frac{31}{4}$ , $y = 5$ , $z = 0$ , and $w = 0$ . This means:

$\frac{31}{4}\vec{a} + 5\vec{b} = 6 \hat{i} + 9 \hat{j}$

Scenario 2: Setting x = 0 and y = 0

If we set $x = 0$ and $y = 0$ , the system becomes:

$-z + 6w = 6$
$-3z + 2w = 9$

Multiply the first equation by -3:

$3z - 18w = -18$
$-3z + 2w = 9$

Adding the modified first equation to the second equation eliminates $z$ :

$-16w = -9$

$w = \frac{9}{16}$

Substitute $w = \frac{9}{16}$ into the first equation:

$-z + 6(\frac{9}{16}) = 6$

$-z + \frac{27}{8} = 6$

$-z = 6 - \frac{27}{8} = \frac{48 - 27}{8} = \frac{21}{8}$

$z = -\frac{21}{8}$

Thus, another possible solution is $x = 0$ , $y = 0$ , $z = -\frac{21}{8}$ , and $w = \frac{9}{16}$ . This means:

$-\frac{21}{8}\vec{c} + \frac{9}{16}\vec{d} = 6 \hat{i} + 9 \hat{j}$

Scenario 3: Setting w = 0

If we set $w=0$ , the system becomes:

$4x - 5y - z = 6$
$-4x + 8y - 3z = 9$

Adding these two equations, we get:

$3y - 4z = 15$

Let's solve for $y$ :

$y = (15 + 4z)/3 = 5 + (4/3)z$

Now, substitute $y$ back into the first equation:

$4x - 5(5 + (4/3)z) - z = 6$

$4x - 25 - (20/3)z - z = 6$

$4x = 31 + (23/3)z$

$x = 31/4 + (23/12)z$

If we choose a value for $z$ , we can find corresponding values for $x$ and $y$ . For example, if we set $z = 0$ , we get $x = 31/4$ and $y = 5$ , which is the solution we found in Scenario 1.

Choosing the Expression

Given the infinite solutions, the specific expression that produces the resultant vector will depend on the context or any additional constraints provided in the problem. However, the process we have outlined demonstrates how to find a linear combination of vectors that results in a desired vector. By setting up a system of equations and solving for the unknown coefficients, we can express any vector as a combination of other vectors.

In this article, we explored the fundamental concepts of vector operations, specifically focusing on how to find a linear combination of vectors that results in a desired resultant vector. We delved into the operations of vector addition and scalar multiplication, emphasizing their roles in creating linear combinations. We then applied these concepts to a specific problem, where we were tasked with finding coefficients for a set of given vectors that would produce a target vector.

Through the process of setting up and solving a system of linear equations, we demonstrated how to determine the scalars needed to achieve the desired resultant vector. We also highlighted the fact that, in many cases, there may be multiple solutions, and the specific solution chosen may depend on additional constraints or considerations.

The ability to manipulate vectors and find their resultant effects is crucial in numerous fields. From physics and engineering to computer graphics and data analysis, vectors serve as essential tools for representing and manipulating quantities that have both magnitude and direction. Understanding vector operations and how to solve for linear combinations empowers us to analyze complex systems, predict outcomes, and design solutions.

This article serves as a foundation for further exploration into the world of vectors and their applications. Whether you are a student learning the basics or a professional applying these concepts in your work, a solid understanding of vector operations will undoubtedly prove invaluable. As you continue your journey, remember that vectors are not just abstract mathematical entities; they are powerful tools that can help us understand and shape the world around us.