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. In a sketch, show r, F, and the origin.

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Understand the problem: We have a force vector $ \mathbf{F} = (-5.00 \text{ N})\mathbf{i} + (4.00 \text{ N})\mathbf{j} $ and a position vector $ \mathbf{r} = (-0.450 \text{ m})\mathbf{i} + (0.150 \text{ m})\mathbf{j} $. We need to sketch these vectors starting from the origin.

Draw the coordinate axes: Begin by drawing a standard Cartesian coordinate system with the x-axis and y-axis intersecting at the origin (0,0).

Plot the position vector $ \mathbf{r} $: From the origin, move 0.450 meters in the negative x-direction (left) and 0.150 meters in the positive y-direction (up). Mark this point and draw an arrow from the origin to this point to represent $ \mathbf{r} $.

Plot the force vector $ \mathbf{F} $: From the origin, move 5.00 N in the negative x-direction (left) and 4.00 N in the positive y-direction (up). Draw an arrow from the origin to this point to represent $ \mathbf{F} $.

Label the vectors: Clearly label the position vector as $ \mathbf{r} $ and the force vector as $ \mathbf{F} $ on your sketch. Ensure the origin is marked as well.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Representation

Vectors are mathematical entities used to represent quantities that have both magnitude and direction, such as force and position. In this problem, the force vector F and position vector r are expressed in terms of their components along the i (x-axis) and j (y-axis) directions, which helps in visualizing and calculating their effects in a two-dimensional plane.

Vector Addition and Subtraction

Vector addition and subtraction are operations that combine or resolve vectors into resultant vectors. This concept is crucial for understanding how different forces and positions interact in a system. In the given problem, understanding how to graphically represent and manipulate these vectors is essential for sketching the scenario accurately.

Coordinate System

A coordinate system is a framework used to define the position of points in space, typically using axes like the x-axis and y-axis. In this problem, the origin is the reference point from which the position vector r is measured, and it is essential to understand how vectors are positioned relative to this origin to accurately sketch the scenario.

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Related Practice

Textbook Question

Calculate the torque (magnitude and direction) about point O due to the force F in each of the cases sketched in Fig. E10.1. In each case, both the force F and the rod lie in the plane of the page, the rod has length 4.00 m, and the force has magnitude F = 10.0 N.

Textbook Question

A metal bar is in the $xy$ -plane with one end of the bar at the origin. A force $$\overrightarrow{F}$=$\left$(7.00N$\right$)i+(-3.00N)j$ is applied to the bar at the point $x=3.00$\text{ m}$$ , $y=4.00$\text{ m}$$ . What are the magnitude and direction of the torque with respect to the origin produced by $\vec{F}$ ?

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Textbook Question

A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long, and he exerts a 17.0-N force at the end of the handle at 37° with the handle (Fig. E10.7). What torque does the machinist exert about the center of the nut?

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Textbook Question

A metal bar is in the $x y$ -plane with one end of the bar at the origin. A force $\vec{F} = (7.00 N) i + (- 3.00 N) j$ is applied to the bar at the point $x equals 3.00 m$ , $y equals 4.00 m$ . In terms of unit vectors $i$ and $j$ , what is the position vector $r$ for the point where the force is applied?

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