Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{F}(x, y)=4 x \mathbf{i}-y^{2} \mathbf{j}\) and \((x, y)\) is on the positive \(y\) -axis, then the vector points in the negative \(y\) -direction.

Short answer

The statement is true.

Step-by-step solution

Substitute the x-value

The first step is to substitute the x coordinate as 0 in \( \mathbf{F}(x, y)\) because any point on the y-axis has x-coordinate as 0. So, the vector becomes \( \mathbf{F}(0, y)=0\mathbf{i}-y^2\mathbf{j} \). This simplifies to \( \mathbf{F}(0, y)=-y^2\mathbf{j} \).

Determine the direction

Next, analyze the direction of the vector. On the positive y-axis, \( y^2 \)and thus \( -y^2 \) will be always negative. This means that j-component (vertical component) of \( \mathbf{F}(0, y) \) is negative, which points in the negative y-direction.

Validate the statement

Finally, compare the direction of \(\mathbf{F}(0, y)\) with the claim in the exercise, which says that the vector points in the negative y-direction when the point is on the positive y-axis. The claim matches our solution. Thus, the statement is true.

Key concepts

True or False Problems

True or false problems are a type of practice that presents a statement and requires students to assess its validity. The objective is to promote critical thinking by forcing one to evaluate whether the provided information is accurate or not. These problems are particularly effective in subjects like vector calculus, where understanding the relationships between equations and geometric representations is crucial.

When solving true or false problems, it's essential for students to not only know the concepts but also to substantiate their answer with a proper explanation or counterexample if the statement is false. This requires a deep understanding of the subject matter to ensure that the evaluation is accurate. It’s not just about guessing but about applying knowledge to validate or debunk the given statement, which is a valuable skill in problem-solving and analysis.

Vector Direction

Vector direction is the orientation of a vector in space, described by the components of the vector along different axes, such as the x, y, and z axes in three-dimensional space. In the context of vector calculus, comprehending vector direction is indispensable for analyzing physical quantities like force, velocity, or electric field, which all have both magnitude and direction.

In the given exercise, the direction of a vector is evaluated based on its components. The negative or positive signs of these components indicate the direction of the vector along the corresponding axis. For instance, a negative y-component suggests that the vector points in the negative y-direction. Understanding and visualizing the physical implications of vector components is a fundamental skill for any student tackling vector calculus.

Coordinate Substitution

Coordinate substitution is a mathematical technique used in various contexts, including solving equations and evaluating functions. Specifically, in vector calculus, it involves replacing the variables with specific values that pertain to the point of interest, which can reveal important information about the behavior of a vector field at that point.

In our exercise, substituting the x-coordinate with zero simplifies the vector function and allows us to analyze the behavior of the vector along the y-axis. As students tackle such problems, they need to understand how to apply coordinate substitution correctly and interpret its results within the context of the problem. This is a critical step in simplifying complex problems and can yield insights into the properties of vector fields across various domains.