One force acting on a machine part is F = (-5.00 N)i + (4.00 N)j. The vector from the origin to the point where the force is applied is r = (-0.450 m)i +(0.150 m)j. (a) In a sketch, show r, F, and the origin.

Verified step by step guidance

1. Identify the vectors given in the problem: the force vector \( \mathbf{F} = (-5.00 \, \text{N})\mathbf{i} + (4.00 \, \text{N})\mathbf{j} \) and the position vector \( \mathbf{r} = (-0.450 \, \text{m})\mathbf{i} + (0.150 \, \text{m})\mathbf{j} \).
2. Draw a coordinate system with x and y axes. The origin (0,0) is your starting point for plotting vectors.
3. Plot the position vector \( \mathbf{r} \) starting from the origin. Move -0.450 m along the x-axis (leftward) and then 0.150 m along the y-axis (upward). Mark the endpoint as the point where the force is applied.
4. From the endpoint of the position vector \( \mathbf{r} \), draw the force vector \( \mathbf{F} \). Move -5.00 N along the x-axis (leftward) and 4.00 N along the y-axis (upward) from the endpoint of \( \mathbf{r} \).
5. Label the vectors appropriately and ensure the sketch clearly shows the origin, the vectors \( \mathbf{r} \) and \( \mathbf{F} \), and the directions they are pointing.

Key Concepts

Vector Representation

Vectors are mathematical entities that have both magnitude and direction. In this context, the force vector F and the position vector r are represented in a Cartesian coordinate system, where 'i' and 'j' denote the unit vectors in the x and y directions, respectively. Understanding how to graphically represent these vectors is essential for visualizing their relationships and effects.

Force and Position Vectors

The force vector F describes the influence acting on an object, while the position vector r indicates the location where this force is applied relative to a reference point, typically the origin. The components of these vectors can be used to analyze the system's behavior, such as calculating torque or determining the resultant force acting on the object.

Sketching Vectors

Creating a sketch of vectors involves plotting them on a coordinate plane, starting from the origin. The position vector r is drawn from the origin to the point (-0.450 m, 0.150 m), while the force vector F is represented starting from the same point, illustrating its direction and magnitude. This visual representation aids in understanding the spatial relationship between the force and the point of application.