How do you calculate a displacement vector?

To calculate a displacement vector, you need to know the initial position and the final position of the object in question. The displacement vector is a vector that points from the initial position to the final position and represents the change in position of the object.

Let's denote the initial position vector as $\vec{r_i}$ and the final position vector as $\vec{r_f}$. The displacement vector, which we'll call $\vec{d}$, is found by subtracting the initial position vector from the final position vector:

$$\vec{d} = \vec{r_f} - \vec{r_i}$$

Here's a step-by-step guide to calculating the displacement vector:

1. Identify Initial and Final Positions: Determine the coordinates of the initial and final positions. Let's say the initial position is given by the coordinates $(x_i, y_i, z_i)$ and the final position is given by the coordinates $(x_f, y_f, z_f)$ in a three-dimensional space.

2. Write Position Vectors: Write the position vectors for the initial and final positions using the coordinates:
$$\vec{r_i} = x_i\hat{i} + y_i\hat{j} + z_i\hat{k}$$
$$\vec{r_f} = x_f\hat{i} + y_f\hat{j} + z_f\hat{k}$$
Here, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are the unit vectors in the direction of the x, y, and z axes, respectively.

3. Subtract the Initial Position Vector from the Final Position Vector: Perform the vector subtraction to find the displacement vector:
$$\vec{d} = (x_f\hat{i} + y_f\hat{j} + z_f\hat{k}) - (x_i\hat{i} + y_i\hat{j} + z_i\hat{k})$$

4. Simplify the Expression: Combine like terms to simplify the expression for the displacement vector:
$$\vec{d} = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j} + (z_f - z_i)\hat{k}$$

5. Calculate the Components of the Displacement Vector: Calculate the differences in the x, y, and z components to find the components of the displacement vector:
$$\vec{d} = \Delta x\hat{i} + \Delta y\hat{j} + \Delta z\hat{k}$$
where $\Delta x = x_f - x_i$, $\Delta y = y_f - y_i$, and $\Delta z = z_f - z_i$.

6. Write the Final Displacement Vector: The final displacement vector is now expressed in terms of its components along the x, y, and z axes:
$$\vec{d} = \Delta x\hat{i} + \Delta y\hat{j} + \Delta z\hat{k}$$

This displacement vector $\vec{d}$ gives you both the direction and magnitude of the displacement. The magnitude (or length) of the displacement vector can be found using the Pythagorean theorem in three dimensions:

$$|\vec{d}| = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}$$

This magnitude represents the straight-line distance between the initial and final positions, while the vector itself indicates the direction of that straight line.