91Ó°ÊÓ

Vector addition is a fundamental concept in mathematics and physics, representing the process of combining two or more vectors to form a resultant vector. In our exercise, when vector \( \vec{B} \) is added to vector \( \vec{A} \), the resultant vector is given by \( 6.0 \hat{\mathrm{i}} + 1.0 \hat{\mathrm{j}} \). Understanding vector addition involves several key points:

• The addition of vectors is commutative, meaning \( \vec{A} + \vec{B} = \vec{B} + \vec{A} \).
• Adding vectors involves summing their corresponding components. For example, the \( i \)-components and \( j \)-components are each summed independently.
• The resultant vector gives a new direction and magnitude, derived from the combination of the original vectors.

In our step-by-step solution, vector addition is used to combine \( \vec{A} + \vec{B} \) which results in \( 6.0 \hat{\mathrm{i}} + 1.0 \hat{\mathrm{j}} \). This is a straightforward application of adding the \( i \) and \( j \) components separately. Understanding how to visually add vectors can help as well. You can think of vectors as arrows showing direction and magnitude, placing them tail-to-head to find the resultant vector easily.

Vector subtraction is another crucial concept, which modifies the direction of the vector being subtracted. For instance, subtracting\( \vec{B} \) from \( \vec{A} \) effectively means adding \(-\vec{B}\) (the negation of \( \vec{B} \)) to \( \vec{A} \). In our problem, this operation gives the result \(-4.0 \hat{\mathrm{i}} + 7.0 \hat{\mathrm{j}} \).When performing vector subtraction, keep in mind:

• Vector subtraction is not commutative, which means \( \vec{A} - \vec{B} eq \vec{B} - \vec{A} \).
• Vector subtraction is akin to vector addition of a negative vector. This involves flipping the direction of the vector that is being subtracted.
• Like addition, subtraction involves dealing with corresponding components separately, i.e., \(i\) and \(j\) components.

To solve the exercise, the equations \( \vec{A} + \vec{B} = 6.0 \hat{\mathrm{i}} + 1.0 \hat{\mathrm{j}} \) and \( \vec{A} - \vec{B} = -4.0 \hat{\mathrm{i}} + 7.0 \hat{\mathrm{j}} \) were used together. Solving these equations allows us to find the original vector components of \( \vec{A} \). Understanding vector subtraction well is crucial for solving such exercises easily.

Vectors in two dimensions can be broken down into their components, which simplify calculations involving them. These components are typically aligned along the Cartesian coordinate axes, such as \( \hat{\mathrm{i}} \) and \( \hat{\mathrm{j}} \) for the \( x \) and \( y \) axes respectively.Here are a few important facts about vector components:

• Each vector can be represented as the sum of its components, such as \( \vec{A} = A_x \hat{\mathrm{i}} + A_y \hat{\mathrm{j}} \).
• Components are scalar values which can be added, subtracted, or subjected to other mathematical operations independently.
• When solving vector equations, determining the components often leads to solving a system of equations to find unknown quantities.

In our problem, once the components of \( \vec{A} \) were isolated following both addition and subtraction of \( \vec{B} \), we established that \( \vec{A} = 1.0 \hat{\mathrm{i}} + 4.0 \hat{\mathrm{j}} \). Using these components, we calculated the magnitude of vector \( \vec{A} \) via the formula \( |\vec{A}| = \sqrt{(A_x)^2 + (A_y)^2} \). Understanding vector components is instrumental in converting practical vector problems into manageable mathematical scenarios.