Problem 106 - Fundamentals of Physics

Question

The position vector for a proton is initially \(\vec{r}=\) \(5.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\) and then later is \(\vec{r}=-2.0 \hat{\mathrm{i}}+6.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\), all in meters. (a) What is the proton's displacement vector, and (b) to what plane is that vector parallel?

Step-by-Step Solution

Answer

(a) Displacement vector is \( -7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} \). (b) It is parallel to the xy-plane.

Step 1: Understand Displacement Vector Concept

A displacement vector represents the change in position of an object. It is obtained by subtracting the initial position vector from the final position vector.

Step 2: Identify Initial and Final Vectors

The initial position vector is \( \vec{r}_{initial} = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \) meters. The final position vector is \( \vec{r}_{final} = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \) meters.

Step 3: Calculate Displacement Vector

To find the displacement vector \( \vec{d} \), subtract the initial vector from the final vector: \[ \vec{d} = \vec{r}_{final} - \vec{r}_{initial} = (-2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}}) - (5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}}). \] Simplifying gives: \( \vec{d} = (-7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}}) \).

Step 4: Analyze the Plane of Displacement Vector

The displacement vector is \( -7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}} \), indicating there is no component along the \( \hat{\mathrm{k}} \) direction (z-axis). Thus, the vector lies entirely in the xy-plane.

Key Concepts

Position Vector

In physics, a position vector is an essential concept for understanding the location of an object in space. It extends from the origin of a coordinate system to the object's position. The position vector represents a point in three-dimensional space using the Cartesian coordinate system, where each component along the x, y, and z axes indicates the object's location relative to the origin.

For example, in our problem, the position vector for a proton at the initial moment is given by \( \vec{r}_{initial} = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \) meters. This means that the proton is 5 meters in the positive x-direction, 6 meters in the negative y-direction, and 2 meters in the positive z-direction. These components help describe exactly where the proton is situated at that time.

Vector Subtraction

The method of vector subtraction is a crucial technique to find the displacement vector, particularly when analyzing changes in position over time. Displacement involves finding the difference between the final and initial position vectors of an object.

In mathematical terms, vector subtraction involves taking the x, y, and z components of the final vector and subtracting the respective components of the initial vector:

• Final \, x-component \(-2.0 \hat{\mathrm{i}} \) minus Initial \, x-component \(5.0 \hat{\mathrm{i}} \) results in \(-7.0 \hat{\mathrm{i}} \)
• Final \, y-component \(6.0 \hat{\mathrm{j}} \) minus Initial \, y-component \(-6.0 \hat{\mathrm{j}} \) results in \(12.0 \hat{\mathrm{j}} \)
• Final \, z-component \(2.0 \hat{\mathrm{k}} \) minus Initial \, z-component \(2.0 \hat{\mathrm{k}} \) results in \(0.0 \hat{\mathrm{k}} \)

Putting it together, the displacement vector is calculated as \(-7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}}\).

XY-Plane

The xy-plane is an important concept in the study of physics and engineering. It is the two-dimensional plane formed by the x-axis and y-axis in a Cartesian coordinate system. When vectors or other geometrical entities lie within this plane, they have no component in the z-axis direction.

In the context of our problem, the displacement vector \(-7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}}\) has no z-component. Because the \(\hat{\mathrm{k}}\) component is zero, it means the vector is entirely within the xy-plane. This simplifies analyses and calculations because it reduces the problem from three dimensions to two.

Physics Problem

Solving physics problems involving vectors often requires understanding how objects move and change position over time. Vectors provide a powerful tool because they encapsulate both the magnitude and direction of physical quantities like displacement, velocity, and force.

When tackling a problem like this one, identify and write down known values, such as the initial and final position vectors. Then, apply vector operations like addition, subtraction, and scalar multiplication to solve for unknowns. Make sure to check for symmetries, like whether vectors lie in specific planes.

By carefully working through each step, the seemingly complex problem of determining where and how an object moves becomes clearer and easier to manage. This systematic approach is a cornerstone of physics problem-solving.