91Ó°ÊÓ

Vectors are fundamental in understanding many physical phenomena. A vector represents both a magnitude and a direction. Often, a vector is broken down into simpler parts called components. This makes it easier to handle mathematically.

Consider a vector described in a two-dimensional plane by components along the axes. In our exercise, vectors are expressed as combinations of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which point in the x and y directions respectively.

• \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \): Here, the components are \(-4\) in the x-direction and \(3\) in the y-direction.
• \( \mathbf{b} = 2 \mathbf{i} - \mathbf{j} \): This vector has components \(2\) in the x-direction and \(-1\) in the y-direction.
• \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \): \(6\) is the x-component and \(-7\) is the y-component.

By breaking vectors into components, we can simplify the problem to solving for scalars like \( k \) and \( m \), as in the given exercise. This decomposition leads to a system of linear equations that can be solved individually for each component direction.

Scalar multiplication involves multiplying a vector by a scalar, which is a real number. This operation stretches or shrinks the vector. For example, multiplying the vector \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \) by \( k \) gives a new vector \( k \mathbf{a} = k(-4 \mathbf{i} + 3 \mathbf{j}) = -4k \mathbf{i} + 3k \mathbf{j} \).

The result is:

• The vector's direction remains the same unless the scalar is negative, which reverses the direction.
• The vector's magnitude changes proportionally to the absolute value of the scalar.

In the exercise, scalar multiplication is necessary to express \( \mathbf{r} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \). The goal is to solve for the scalars \( k \) and \( m \) such that the combination matches \( \mathbf{r} \). This requires accurate manipulation and understanding of how scalars interact with vector components.

A linear combination refers to an expression constructed from a set of terms by multiplying each term by a constant and adding the results. In vectors, this means combining vectors using scalar multiplication to find another vector.

For instance, our goal is to express the vector \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \) as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \). Simply put, we want to find scalars \( k \) and \( m \) such that:

• \( \mathbf{r} = k \mathbf{a} + m \mathbf{b} = k(-4 \mathbf{i} + 3 \mathbf{j}) + m(2 \mathbf{i} - \mathbf{j}) \)

Separately equating the components of \( \mathbf{r} \) with those of \( k \mathbf{a} + m \mathbf{b} \) leads to a system of equations, namely \( 6 = -4k + 2m \) and \( -7 = 3k - m \).

Solving these equations determines the precise scalars needed to accurately represent \( \mathbf{r} \) using \( \mathbf{a} \) and \( \mathbf{b} \). This practice is crucial in various fields such as physics and engineering, where expressing complex systems as linear combinations simplifies analysis and problem-solving.