Determine the displacement vector d.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Determine the displacement vector $\vec{d}$.
The initial position is $P_1(4, 5, -3)$ and the final position is $P_2(3, 6, 2)$.
The displacement vector is given by $\vec{d} = \vec{P_2} - \vec{P_1}$.
$$\vec{d} = (3\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}) - (4\mathbf{i} + 5\mathbf{j} - 3\mathbf{k})$$
$$\vec{d} = (3-4)\mathbf{i} + (6-5)\mathbf{j} + (2-(-3))\mathbf{k}$$
$$\vec{d} = -1\mathbf{i} + 1\mathbf{j} + 5\mathbf{k} \text{ m}$$
Step 2: Calculate the work done $W$.
The force vector is $\vec{F} = 5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$ N.
The work done by a constant force is the dot product of the force vector and the displacement vector: $W = \vec{F} \cdot \vec{d}$.
$$W = (5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \cdot (-1\mathbf{i} + 1\mathbf{j} + 5\mathbf{k})$$
$$W = (5)(-1) + (3)(1) + (-2)(5)$$
$$W = -5 + 3 - 10$$
$$W = -2 - 10$$
$$W = -12 \text{ J}$$
The work done on the body is $-12 \text{ J}$.
$$\boxed{-12 \text{ J}}$$
That's 2 down. 3 left today — send the next one.
