Chapter 3

What are (a) the \(x\) component and (b) the \(y\) component of a vector \(\vec{a}\) in the \(x y\) plane if its direction is \(250^{\circ} \mathrm{y}\) counterclockwise from the positive direction of the \(x\) axis and its magnitude is \(7.3 \mathrm{~m}\) ?

A displacement vector \(\vec{r}\) in the \(x y\) plane is \(15 \mathrm{~m}\) long and directed at angle \(\theta=30^{\circ}\) in Fig. 3 -26. Determine (a) the \(x\) component and (b) the \(y\) component of the vector.

The \(x\) component of vector \(\vec{A}\) is \(-25.0 \mathrm{~m}\) and the \(y\) component is \(+40.0 \mathrm{~m}\). (a) What is the magnitude of \(\vec{A} ?\) (b) What is the angle between the direction of \(\vec{A}\) and the positive direction of \(x ?\)

Express the following angles in radians: (a) \(20.0^{\circ}\), (b) \(50.0^{\circ}\), (c) \(100^{\circ}\). Convert the following angles to degrees: (d) \(0.330\) rad, (e) \(2.10 \mathrm{rad},(\mathrm{f}) 7.70 \mathrm{rad}\).

A ship sets out to sail to a point \(120 \mathrm{~km}\) due north. An unexpected storm blows the ship to a point \(100 \mathrm{~km}\) due east of its starting point. (a) How far and (b) in what direction must it now sail to reach its original destination?

Consider two displacements, one of magnitude \(3 \mathrm{~m}\) and another of magnitude \(4 \mathrm{~m}\). Show how the displacement vectors may be combined to get a resultant displacement of magnitude (a) \(7 \mathrm{~m}\), (b) \(1 \mathrm{~m}\), and (c) \(5 \mathrm{~m}\).

A person walks in the following pattern: \(3.1 \mathrm{~km}\) north, then \(2.4 \mathrm{~km}\) west, and finally \(5.2 \mathrm{~km}\) south. (a) Sketch the vector diagram that represents this motion. (b) How far and (c) in what direction would a bird fly in a straight line from the same starting point to the -8 A person walks in the following pattern: \(3.1 \mathrm{~km}\) north, then \(2.4 \mathrm{~km}\) west, and finally \(5.2 \mathrm{~km}\) south. (a) Sketch the vector diagram that represents this motion. (b) How far and (c) in what direction would a bird fly in a straight line from the same starting point to the same final point?

Two vectors are given by and $$\begin{aligned}&\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}}+(1.0 \mathrm{~m}) \hat{\mathrm{k}} \\ &\vec{b}=(-1.0 \mathrm{~m}) \hat{\mathrm{i}}+(1.0 \mathrm{~m}) \hat{\mathrm{j}}+(4.0 \mathrm{~m}) \hat{\mathrm{k}}\end{aligned}$$ In unit-vector notation, find (a) \(\vec{a}+\vec{b}\), (b) \(\vec{a}-\vec{b}\), and \((\mathrm{c})\) a third vector \(\vec{c}\) such that \(\vec{a}-\vec{b}+\vec{c}=0\)

Find the (a) \(x\), (b) \(y\), and (c) \(z\) components of the sum \(\vec{r}\) of the displacements \(\vec{c}\) and \(\vec{d}\) whose components in meters are \(c_{x}=7.4, c_{y}=-3.8, c_{z}=-6.1 ; d_{x}=4.4, d_{y}=-2.0, d_{z}=3.3\) \(-11\) ssM (a) In unit-vector notation, what is the sum \(\vec{a}+\vec{b}\) if \(\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{j}} ?\) What are the (b) magnitude and (c) direction of \(\vec{a}+\vec{b}\) ?

(a) In unit-vector notation, what is the sum \(\vec{a}+\vec{b}\) if \(\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{j}} ? \quad\) What are the (b) magnitude and (c) direction of \(\vec{a}+\vec{b}\) ?

A car is driven east for a distance of \(50 \mathrm{~km}\), then north for 30 \(\mathrm{km}\), and then in a direction \(30^{\circ}\) east of north for \(25 \mathrm{~km}\). Sketch the vector diagram and determine (a) the magnitude and (b) the angle of the car's total displacement from its starting point.

A person desires to reach a point that is \(3.40 \mathrm{~km}\) from her present location and in a direction that is \(35.0^{\circ}\) north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

You are to make four straight-line moves over a flat desert floor, starting at the origin of an \(x y\) coordinate system and ending at the \(x y\) coordinates \((-140 \mathrm{~m}, 30 \mathrm{~m})\). The \(x\) component and \(y\) component of your moves are the following, respectively, in meters: \((20\) and 60\()\), then \(\left(b_{x}\right.\) and \(\left.-70\right)\), then \(\left(-20\right.\) and \(\left.c_{y}\right)\), then \((-60\) and \(-70\) ). What are (a) component \(b_{x}\) and (b) component \(c_{y}\) ? What are (c) the magnitude and (d) the angle (relative to the positive direction of the \(x\) axis) of the overall displacement?

For the displacement vectors \(\vec{a}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \hat{\mathrm{j}}\) and \(\vec{b}=\) \((5.0 \mathrm{~m}) \hat{\mathrm{i}}+(-2.0 \mathrm{~m}) \hat{\mathrm{j}}\), give \(\vec{a}+\vec{b}\) in (a) unit-vector notation, and as (b) a magnitude and (c) an angle (relative to \(\hat{1}\) ). Now give \(\vec{b}-\vec{a}\) in (d) unit-vector notation, and as (e) a magnitude and (f) an angle. -

Three vectors \(\vec{a}, \vec{b}\), and \(\vec{c}\) each have a magnitude of \(50 \mathrm{~m}\) and lie in an \(x y\) plane. Their directions relative to the positive direction of the \(x\) axis are \(30^{\circ}, 195^{\circ}\), and \(315^{\circ}\), respectively. What are (a) the magnitude and (b) the angle of the vector \(\vec{a}+\vec{b}+\vec{c}\), and (c) the magnitude and (d) the angle of \(\vec{a}-\vec{b}+\vec{c}\) ? What are the (e) magnitude and (f) angle of a fourth vector \(\vec{d}\) such that \((\vec{a}+\vec{b})-(\vec{c}+\vec{d})=0 ?\)

In the sum \(\vec{A}+\vec{B}=\vec{C}\), vector \(\vec{A}\) has a magnitude of \(12.0 \mathrm{~m}\) and is angled \(40.0^{\circ}\) counterclockwise from the \(+x\) direction, and vector \(\vec{C}\) has a magnitude of \(15.0 \mathrm{~m}\) and is angled \(20.0^{\circ}\) counterclockwise from the \(-x\) direction. What are (a) the magnitude and (b) the angle (relative to \(+x\) ) of \(\vec{B}\) ?

In a game of lawn chess, where pieces are moved between the centers of squares that are each \(1.00 \mathrm{~m}\) on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for \(5.6 \mathrm{~km}\), but when the snow clears, he discovers that he actually traveled \(7.8 \mathrm{~km}\) at \(50^{\circ}\) north of due east. (a) How far and (b) in what direction must he now travel to reach base camp?

An ant, crazed by the Sun on a hot Texas afternoon, darts over an \(x y\) plane scratched in the dirt. The \(x\) and \(y\) components of four consecutive darts are the following, all in centimeters: \((30.0,\), 40.0), \(\left(b_{x},-70.0\right),\left(-20.0, c_{y}\right),(-80.0,-70.0)\). The overall displacement of the four darts has the \(x y\) components \((-140,-20.0)\). What are (a) \(b_{x}\) and (b) \(c_{y} ?\) What are the (c) magnitude and (d) angle (relative to the positive direction of the \(x\) axis) of the overall displacement?

(a) What is the sum of the following four vectors in unitvector notation? For that sum, what are (b) the magnitude, (c) the angle in degrees, and (d) the angle in radians? $$\begin{array}{ll}\vec{E}: 6.00 \mathrm{~m} \mathrm{at}+0.900 \mathrm{rad} & \vec{F}: 5.00 \mathrm{~m} \text { at }-75.0^{\circ} \\\\\vec{G}: 4.00 \mathrm{~m} \mathrm{at}+1.20 \mathrm{rad} & \vec{H}: 6.00 \mathrm{~m} \text { at }-210^{\circ}\end{array}$$

If \(\vec{B}\) is added to \(\vec{C}=3.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}\), the result is a vector in the positive direction of the \(y\) axis, with a magnitude equal to that of \(\vec{C}\). What is the magnitude of \(\vec{B}\) ?

Vector \(\vec{A}\), which is directed along an \(x\) axis, is to be added to vector \(\vec{B}\), which has a magnitude of \(7.0 \mathrm{~m}\). The sum is a third vector that is directed along the \(y\) axis, with a magnitude that is \(3.0\) times that of \(\vec{A}\). What is that magnitude of \(\vec{A}\) ?

Oasis \(B\) is \(25 \mathrm{~km}\) due east of oasis \(A\). Starting from oasis \(A\), a camel walks \(24 \mathrm{~km}\) in a direction \(15^{\circ}\) south of east and then walks \(8.0 \mathrm{~km}\) due north. How far is the camel then from oasis \(B\) ?

What is the sum of the following four vectors in (a) unitvector notation, and as (b) a magnitude and (c) an angle? $$\begin{array}{ll}\vec{A}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}+(3.00 \mathrm{~m}) \hat{\mathrm{j}} & \vec{B}: 4.00 \mathrm{~m}, \text { at }+65.0^{\circ} \\ \vec{C}=(-4.00 \mathrm{~m}) \hat{\mathrm{i}}+(-6.00 \mathrm{~m}) \hat{\mathrm{j}} & \vec{D}: 5.00 \mathrm{~m}, \text { at }-235^{\circ}\end{array}$$

If \(\vec{d}_{1}+\vec{d}_{2}=5 \vec{d}_{3}, \vec{d}_{1}-\vec{d}_{2}=3 \vec{d}_{3}\), and \(\vec{d}_{3}=2 \hat{i}+4 \hat{j}\), then what are, in unit-vector notation, (a) \(\vec{d}_{1}\) and (b) \(\vec{d}_{2}\) ?

Two beetles run across flat sand, starting at the same point. Beetle 1 runs \(0.50 \mathrm{~m}\) due east, then \(0.80 \mathrm{~m}\) at \(30^{\circ}\) north of due east. Beetle 2 also makes two runs; the first is \(1.6 \mathrm{~m}\) at \(40^{\circ}\) east of due north. What must be (a) the magnitude and (b) the direction of its second run if it is to end up at the new location of beetle \(1 ?\)

Here are two vectors: $$\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(3.0 \mathrm{~m}) \hat{\mathrm{j}} \text { and } \vec{b}=(6.0 \mathrm{~m}) \hat{\mathrm{i}}+(8.0 \mathrm{~m}) \hat{\mathrm{j}}$$ What are (a) the magnitude and (b) the angle (relative to \(\hat{1}\) ) of \(\vec{a}\) ? What are (c) the magnitude and (d) the angle of \(\vec{b}\) ? What are (e) the magnitude and (f) the angle of \(\vec{a}+\vec{b} ;(\mathrm{g})\) the magnitude and (h) the angle of \(\vec{b}-\vec{a} ;\) and (i) the magnitude and (j) the angle of \(\vec{a}-\vec{b} ?(\mathrm{k})\) What is the angle between the directions of \(\vec{b}-\vec{a}\) and \(\vec{a}-\vec{b} ?\)

Two vectors, \(\vec{r}\) and \(\vec{s}\), lie in the \(x y\) plane. Their magnitudes are \(4.50\) and \(7.30\) units, respectively, and their directions are \(320^{\circ}\) and \(85.0^{\circ}\), respectively, as measured counterclockwise from the positive \(x\) axis. What are the values of (a) \(\vec{r} \cdot \vec{s}\) and (b) \(\vec{r} \times \vec{s}\) ?

If \(\vec{d}_{1}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) and \(\vec{d}_{2}=-5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\), then what is \(\left(\vec{d}_{1}+\vec{d}_{2}\right) \cdot\left(\vec{d}_{1} \times 4 \vec{d}_{2}\right) ?\)

Three vectors are given by \(\vec{a}=3.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}\), \(\vec{b}=-1.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\), and \(\vec{c}=2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+1.0 \hat{\mathrm{k}}\). Find (a) \(\vec{a} \cdot(\vec{b} \times \vec{c})\), (b) \(\vec{a} \cdot(\vec{b}+\vec{c})\), and \((\mathrm{c}) \vec{a} \times(\vec{b}+\vec{c})\)

For the following three vectors, what is \(3 \vec{C} \cdot(2 \vec{A} \times \vec{B}) ?\) $$\begin{aligned} &\vec{A}=2.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}-4.00 \hat{\mathrm{k}} \\ &\vec{B}=-3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}} \quad \vec{C}=7.00 \hat{\mathrm{i}}-8.00 \hat{\mathrm{j}}\end{aligned}$$

Vector \(\vec{A}\) has a magnitude of \(6.00\) units, vector \(\vec{B}\) has a mag. nitude of \(7.00\) units, and \(\vec{A} \cdot \vec{B}\) has a value of \(14.0\). What is the angle between the directions of \(\vec{A}\) and \(\vec{B}\) ?

Displacement \(\vec{d}_{1}\) is in the \(y z\) plane \(63.0^{\circ}\) from the positive direction of the \(y\) axis, has a positive \(z\) component, and has a magnitude of \(4.50 \mathrm{~m}\). Displacement \(\vec{d}_{2}\) is in the \(x z\) plane \(30.0^{\circ}\) from the positive direction of the \(x\) axis, has a positive \(z\) component, and has magnitude \(1.40 \mathrm{~m}\). What are (a) \(\vec{d}_{1} \cdot \vec{d}_{2}\), (b) \(\vec{d}_{1} \times \vec{d}_{2}\), and (c) the angle between \(\vec{d}_{1}\) and \(\vec{d}_{2}\) ?

Use the definition of scalar product, \(\vec{a} \cdot \vec{b}=a b \cos \theta\), and the fact that \(\vec{a} \cdot \vec{b}=a_{x} b_{x}+a_{y} b_{y}+a_{z} b_{z}\) to cal- culate the angle between the two vectors given by \(\vec{a}=3.0 \hat{\mathrm{i}}+\) \(3.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}\) and \(\vec{b}=2.0 \hat{\mathrm{i}}+1.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}\)

In the product \(\vec{F}=q \vec{v} \times \vec{B}\), take \(q=2\), $$\vec{v}=2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}} \text { and } \vec{F}=4.0 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}$$ What then is \(\vec{B}\) in unit-vector notation if \(B_{x}=B_{y} ?\)

Vector \(\vec{a}\) has a magnitude of \(5.0 \mathrm{~m}\) and is directed east. Vector \(\vec{b}\) has a magnitude of \(4.0 \mathrm{~m}\) and is directed \(35^{\circ}\) west of due north. What are (a) the magnitude and (b) the direction of \(\vec{a}+\vec{b}\) ? What are (c) the magnitude and (d) the direction of \(\vec{b}-\vec{a} ?\) (e) Draw a vector diagram for each combination.

Vectors \(\vec{A}\) and \(\vec{B}\) lie in an \(x y\) plane. \(\vec{A}\) has magnitude \(8.00\) and angle \(130^{\circ} ; \vec{B}\) has components \(B_{x}=-7.72\) and \(B_{y}=-9.20\). What are the angles between the negative direction of the \(y\) axis and (a) the direction of \(\vec{A}\), (b) the direction of the product \(\vec{A} \times \vec{B}\), and \((\mathrm{c})\) the direction of \(\vec{A} \times(\vec{B}+3.00 \hat{\mathrm{k}}) ?\)

Two vectors \(\vec{a}\) and \(\vec{b}\) have the components, in meters, \(a_{x}=3.2, a_{y}=1.6, b_{x}=0.50, b_{y}=4.5 .\) (a) Find the angle between the directions of \(\vec{a}\) and \(\vec{b}\). There are two vectors in the \(x y\) plane that are perpendicular to \(\vec{a}\) and have a magnitude of \(5.0 \mathrm{~m} .\) One, vector \(\vec{c}\), has a positive \(x\) component and the other, vector \(\vec{d}\), a negative \(x\) component. What are (b) the \(x\) component and (c) the \(y\) component of vector \(\vec{c}\), and (d) the \(x\) component and (e) the \(y\) component of vector \(\vec{d}\) ?

A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, \(90.0 \mathrm{~km}\) due north. The sailor, however, ends up \(50.0 \mathrm{~km}\) due east of the starting point. (a) How far and (b) in what direction must the sailor now sail to reach the original destination?

Vector \(\vec{d}_{1}\) is in the negative direction of a \(y\) axis, and vector \(\vec{d}_{2}\) is in the positive direction of an \(x\) axis. What are the directions of (a) \(\vec{d}_{2} / 4\) and (b) \(\vec{d}_{1} /(-4) ?\) What are the magnitudes of products (c) \(\vec{d}_{1} \cdot \vec{d}_{2}\) and (d) \(\vec{d}_{1} \cdot\left(\vec{d}_{2} / 4\right) ?\) What is the direction of the vector resulting from (e) \(\vec{d}_{1} \times \vec{d}_{2}\) and (f) \(\vec{d}_{2} \times \vec{d}_{1}\) ? What is the magnitude of the vector product in (g) part (e) and (h) part (f)? What are the (i) magnitude and (j) direction of \(\vec{d}_{1} \times\left(\vec{d}_{2} / 4\right)\) ?

Rock faults are ruptures along which opposite faces of rock have slid past each other. In Fig. \(3-35\), points \(A\) and \(B\) coincided before the rock in the foreground slid down to the right. The net displacement \(\overrightarrow{A B}\) is along the plane of the fault. The horizontal component of \(\overrightarrow{A B}\) is the strike-slip \(A C\). The component of \(\overrightarrow{A B}\) that is directed down the plane of the fault is the dip-slip \(A D\). (a) What is the magnitude of the net displacement \(\overrightarrow{A B}\) if the strike-slip is \(22.0 \mathrm{~m}\) and the dip- slip is \(17.0 \mathrm{~m} ?\) (b) If the plane of the fault is inclined at angle \(\phi=52.0^{\circ}\) to the horizontal, what is the vertical component of \(\overrightarrow{A B}\) ?

A vector \(\vec{a}\) of magnitude 10 units and another vector \(\vec{b}\) of magnitude \(6.0\) units differ in directions by \(60^{\circ} .\) Find (a) the scalar product of the two vectors and (b) the magnitude of the vector product \(\vec{a} \times \vec{b}\)

A particle undergoes three successive displacements in plane, as follows: \(\vec{d}_{1}, 4.00 \mathrm{~m}\) southwest; then \(\vec{d}_{2}, 5.00 \mathrm{~m}\) east; an finally \(\vec{d}_{3}, 6.00 \mathrm{~m}\) in a direction \(60.0^{\circ}\) north of east. Choose a coon dinate system with the \(y\) axis pointing north and the \(x\) axis pointin east. What are (a) the \(x\) component and (b) the \(y\) component of \(\vec{d}_{1}\) What are (c) the \(x\) component and (d) the \(y\) component of \(\vec{d}_{2}\) What are (e) the \(x\) component and (f) the \(y\) component of \(\vec{d}_{3}\) Next, consider the net displacement of the particle for the thre successive displacements. What are (g) the \(x\) component, (h) the component, (i) the magnitude, and (j) the direction of the net dis placement? If the particle is to return directly to the starting poin (k) how far and (1) in what direction should it move?

If \(\vec{B}\) is added to \(\vec{A}\), the result is \(6.0 \hat{\mathrm{i}}+1.0 \hat{\mathrm{j}} .\) If \(\vec{B}\) is subtracted from \(\vec{A}\), the result is \(-4.0 \hat{\mathrm{i}}+7.0 \hat{\mathrm{j}}\). What is the magnitude of \(\vec{A} ?\)

A vector \(\vec{d}\) has a magnitude of \(2.5 \mathrm{~m}\) and points north. What are (a) the magnitude and (b) the direction of \(4.0 \vec{d}\) ? What are (c) the magnitude and (d) the direction of \(-3.0 \vec{d}\) ?

A has the magnitude \(12.0 \mathrm{~m}\) and is angled \(60.0^{\circ}\) counterclockwise from the positive direction of the \(x\) axis of an \(x y\) coordinate system. Also, \(\vec{B}=(12.0 \mathrm{~m}) \hat{\mathrm{i}}+(8.00 \mathrm{~m}) \hat{\mathrm{j}}\) on that same coordinate system. We now rotate the system counterclockwise about the origin by \(20.0^{\circ}\) to form an \(x^{\prime} y^{\prime}\) system. On this new system, what are (a) \(\vec{A}\) and (b) \(\vec{B}\), both in unit-vector notation?

If \(\vec{a}-\vec{b}=2 \vec{c}, \vec{a}+\vec{b}=4 \vec{c}\), and \(\vec{c}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}\), then what are (a) \(\vec{a}\) and \((\mathrm{b}) \vec{b} ?\)

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball \(3.66 \mathrm{~m}\) north, the second \(1.83 \mathrm{~m}\) southeast, and the third \(0.91 \mathrm{~m}\) southwest. What are (a) the magnitude and (b) the direction of the displacement needed to get the ball into the hole on the first putt?

Here are three vectors in meters: $$\begin{aligned}&\vec{d}_{1}=-3.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} \\ &\vec{d}_{2}=-2.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} \\\ &\vec{d}_{3}=2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}+1.0 \hat{\mathrm{k}} \end{aligned}$$ What results from (a) \(\vec{d}_{1} \cdot\left(\vec{d}_{2}+\vec{d}_{3}\right)\), (b) \(\vec{d}_{1} \cdot\left(d_{2} \times \vec{d}_{3}\right)\), and (c) \(\vec{d}_{1} \times\left(\vec{d}_{2}+\vec{d}_{3}\right) ?\)

A room has dimensions \(3.00 \mathrm{~m}\) (height) \(x\) \(3.70 \mathrm{~m} \times 4.30 \mathrm{~m}\). A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Greater? (d) Equal? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. (f) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)