91Ó°ÊÓ

When we talk about scalar multiplication in the context of vectors, we're referring to the process of multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction, at least as long as the scalar is positive. If the scalar is negative, the vector not only changes in magnitude but also reverses its direction.

For example, if you have a vector \( \vec{u} \) and you multiply it by a scalar \( a \), you get a new vector \( \vec{v} = a\vec{u} \). The components of \( \vec{u} \) are scaled by \( a \), so if \( \vec{u} = u_x\vec{i} + u_y\vec{j}\), then \( \vec{v} = a\cdot u_x\vec{i} + a\cdot u_y\vec{j}\).

Understanding scalar multiplication is crucial because it leads us to identify proportional relationships between vectors, a key aspect in determining if two vectors are parallel, as showcased in the textbook exercise.

Vectors in two dimensions have two components: the \( x \) component (usually aligned with the \( \vec{i} \) unit vector) and the \( y \) component (aligned with the \( \vec{j} \) unit vector). The \( \vec{i} \) and \( \vec{j} \) are the standard notation for unit vectors along the \( x \) and \( y \) axes, respectively.

In our textbook problem, the vector \( \vec{v} \) has components \( 7a \vec{i} \) and \( -3\vec{j}\), while \( \vec{w} \) has \( a^2\vec{i} \) and \( 9\vec{j}\). When solving problems involving vector components, we often work with each component independently to determine relationships between vectors or to perform vector addition and subtraction. Accurate handling of these components is essential for understanding vectors' behaviors and their interactions.

Proportional relationships are at the heart of understanding parallel vectors. Two non-zero vectors are parallel if and only if one is a scalar multiple of the other. This means that the corresponding components of two parallel vectors must be proportional. In other words, the division of each corresponding component must yield the same scalar value if the vectors are indeed parallel.

In the exercise provided, we examine the ratios of corresponding components: \( (7a)/(a^2) \) for the \( \vec{i} \) components, and \( -3/9 \) for the \( \vec{j} \) components. If these fractions are equal, it illustrates a proportional relationship between \( \vec{v} \) and \( \vec{w} \) and consequently proves their parallelism. It is this link between proportionality and parallel vectors that we leverage to solve for the scalar \( a \).

The final step in our textbook example involves solving equations to find the specific scalar that produces parallelism. Once we've established the proportional relationship, we set the two ratios equal to each other and solve for the unknown scalar, \( a \).

This step requires algebraic manipulation, including cross-multiplication, as seen when we equate \( 7/a \) (from the \( \vec{i} \) components ratio) to \( -1/3 \) (the simplified ratio of the \( \vec{j} \) components). The equation \( 7/a = -1/3 \) leads us to find that \( a = -21 \) after cross-multiplying and simplifying. This value of \( a \) provides the only solution in this context, making \( \vec{v} \) and \( \vec{w} \) parallel vectors. Mastery of solving equations is thus not only essential in finding such solutions but also broadly applicable across various areas of mathematics.