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Chapter 4: Problem 106

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

Short Answer

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

Step by step solution

01

Understand the Problem

The problem gives us an initial position vector \(\vec{r_1} = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}}\) and a final position vector \(\vec{r_2} = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}}\). We are asked to find the displacement vector and determine which plane it is parallel to.
02

Calculate the Displacement Vector

The displacement vector \(\vec{d}\) is calculated using the formula \(\vec{d} = \vec{r_2} - \vec{r_1}\). Substituting the given vectors, we get:\[\vec{d} = (-2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}}) - (5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}})\]Solving this, we find the displacement vector:\[\vec{d} = (-2.0 - 5.0) \hat{\mathrm{i}} + (6.0 + 6.0) \hat{\mathrm{j}} + (2.0 - 2.0) \hat{\mathrm{k}}\]\[\vec{d} = -7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}}\]
03

Determine the Plane Parallel to the Displacement Vector

The displacement vector \(\vec{d} = -7.0 \hat{\mathrm{i}} + 12.0 \hat{\mathrm{j}} + 0.0 \hat{\mathrm{k}}\) has no component along the \(\hat{\mathrm{k}}\) axis, indicating that it lies in the XY plane. Therefore, the displacement vector is parallel to the XY plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Position Vector
A position vector is a vector that represents the position of a point in space relative to a reference point, often the origin of a coordinate system.
In three-dimensional space, it is often written in terms of unit vectors, \(\hat{i}, \hat{j},\) and \(\hat{k}\). These unit vectors correspond to the x, y, and z axes, respectively.
For example, the position vector \(\vec{r} = 5.0 \hat{i} - 6.0 \hat{j} + 2.0 \hat{k}\) indicates that the point is located 5 units along the x-axis, -6 units along the y-axis, and 2 units along the z-axis from the origin.
  • The components \(5.0, -6.0,\) and \(2.0\) are the respective magnitudes along the \(\hat{i}, \hat{j}, \) and \(\hat{k}\) directions.
  • The position vector can change over time as objects move through space.
    This change is described by another vector called the displacement vector.
Dilucidating the XY Plane
The XY plane is one of the principal planes in three-dimensional Cartesian coordinate systems.
It is defined by the set of all points that have a z-coordinate of zero.
This means that every point on the XY plane can be described solely by x and y coordinates, with no contribution from the z-axis.
  • In the context of vectors, if a vector has a zero component in the \(\hat{k}\) direction, it lies entirely within the XY plane.
  • Examining the displacement vector \(\vec{d} = -7.0 \hat{i} + 12.0 \hat{j} + 0.0 \hat{k}\) shows that it has no z component, confirming that it rests in the XY plane.
Understanding which plane a vector lies in is crucial for solving problems related to motion and forces, as it simplifies calculations by reducing the number of dimensions to consider.
Exploring Vector Components
Vector components are the projections of a vector along the axes of the coordinate system.
They are crucial for breaking down vectors into simpler parts that are easier to work with.
For any given vector \(\vec{v} = a \hat{i} + b \hat{j} + c \hat{k}\), the components \(a, b,\) and \(c\) are the contributions of the vector along the x, y, and z axes, respectively.
  • These components are often used to calculate the resultant vectors when multiple vectors interact.
  • In the displacement vector \(\vec{d} = -7.0 \hat{i} + 12.0 \hat{j} + 0.0 \hat{k}\), the components -7.0 and 12.0 indicate that the total movement is spread over the x and y directions.
By understanding vector components, one can seamlessly navigate through vector operations like addition, subtraction, and finding magnitudes, which are fundamental in physics and engineering.

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Most popular questions from this chapter

Two seconds after being projected from ground level, a projectile is displaced \(40 \mathrm{~m}\) horizontally and \(53 \mathrm{~m}\) vertically above its launch point. What are the (a) horizontal and (b) vertical components of the initial velocity of the projectile? (c) At the instant the projectile achieves its maximum height above ground level, how far is it displaced horizontally from the launch point?

A \(200-\mathrm{m}\) -wide river flows due east at a uniform speed of \(2.0 \mathrm{~m} / \mathrm{s}\). A boat with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) relative to the water leaves the south bank pointed in a direction \(30^{\circ}\) west of north. What are the (a) magnitude and (b) direction of the boat's velocity relative to the ground? (c) How long does the boat take to cross the river?

Oasis \(A\) is \(90 \mathrm{~km}\) due west of oasis \(B .\) A desert camel leaves \(A\) and takes \(50 \mathrm{~h}\) to walk \(75 \mathrm{~km}\) at \(37^{\circ}\) north of due east. Next it takes \(35 \mathrm{~h}\) to walk \(65 \mathrm{~km}\) due south. Then it rests for \(5.0 \mathrm{~h}\). What are the (a) magnitude and (b) direction of the camel's displacement relative to \(A\) at the resting point? From the time the camel leaves \(A\) until the end of the rest period, what are the (c) magnitude and (d) direction of its average velocity and (e) its average speed? The camel's last drink was at \(A ;\) it must be at \(B\) no more than \(120 \mathrm{~h}\) later for its next drink. If it is to reach \(B\) just in time, what must be the (f) magnitude and (g) direction of its average velocity after the rest period?

A boy whirls a stone in a horizontal circle of radius \(1.5 \mathrm{~m}\) and at height \(2.0 \mathrm{~m}\) above level ground. The string breaks, and the stone flies off horizontally and strikes the ground after traveling a horizontal distance of \(10 \mathrm{~m}\). What is the magnitude of the centripetal acceleration of the stone during the circular motion?

A particle \(P\) travels with constant speed on a circle of radius \(r=\) \(3.00 \mathrm{~m}\) (Fig. \(4-56\) ) and completes one revolution in \(20,0 \mathrm{~s}\). The particle passes through \(O\) at time \(t=0 .\) State the following vectors in magnitudeangle notation (angle relative to the positive direction of \(x\) ). With respect to \(O\), find the particle's position vector at the times \(t\) of (a) \(5.00 \mathrm{~s}\), (b) \(7.50 \mathrm{~s}\), and (c) \(10.0 \mathrm{~s}\) (d) For the \(5.00 \mathrm{~s}\) interval from the end of the fifth second to the end of the tenth second, find the particle's displacement. For that interval, find (e) its average velocity and its velocity at the (f) beginning and (g) end. Next, find the acceleration at the (h) beginning and (i) end of that interval.
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