# Calculating work using vectors.

• Random-Hero-
In summary, the work done can be calculated by taking the dot product of the force vector (25N) and the displacement vector (Q-P), with the direction vector (2,3,-1) providing the direction of the force. The force vector is determined by multiplying the magnitude of the force by the unit normal in the direction of the force, which can be calculated from the direction vector.
Random-Hero-

## Homework Statement

Calculate the work done if a 25N force acting in a direction of vector (2,3,-1) moves an object from P(2,-3,1) to Q(5,0,2).

## Homework Equations

Work = Force (dot) Displacement

## The Attempt at a Solution

Well force is 25N. But as for displacement, do I just find the displacement of the points P and Q, but then what's the direction vector (2,3,-1) for? I'm just a bit confused on this.

The displacement is going to be the vector Q-P. The direction vector is what direction the force is pushing on the object... you use the 25N part to determine what the magnitude of the force vector is. Then you take the dot product of that vector and Q-P

The direction vector gives you the direction of the 25N force vector. The force vector is just the magnitude of the force times the unit normal in the direction of the force. You should be able to calculate the unit normal of the direction vector (2,3,-1).

## 1. How do you calculate the work done by a force using vectors?

The work done by a force can be calculated using the formula W = Fcosθd, where F is the magnitude of the force, θ is the angle between the force vector and the displacement vector, and d is the displacement. This formula takes into account both the magnitude and direction of the force.

## 2. What is the difference between scalar and vector quantities when calculating work?

Scalar quantities, such as mass and temperature, have only magnitude and no direction. Vector quantities, such as force and displacement, have both magnitude and direction. When calculating work, vector quantities must be used in order to accurately account for direction.

## 3. Can work be calculated for a system of multiple forces?

Yes, work can be calculated for a system of multiple forces by using vector addition. The total work done by a system of forces is equal to the sum of the work done by each individual force.

## 4. What units are used to measure work?

The unit of work is the joule (J), which is equivalent to a newton-meter (N·m). This unit is derived from the formula for work, where force is measured in newtons and displacement is measured in meters.

## 5. Can work be negative? What does this indicate?

Yes, work can be negative. A negative value for work indicates that the force is acting in the opposite direction of the displacement. This could represent work being done against the force, rather than by the force.

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