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Chapter 11: Problem 10

\((a)\) Calculate \(\overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\), where \(\overrightarrow{\mathbf{a}}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{b}}=-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\), and \(\overrightarrow{\mathbf{c}}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}} .\) (b) Calculate the angle between \(\overrightarrow{\mathbf{r}}\) and the \(+z\) axis. ( \(c\) ) Find the angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\).

Short Answer

Expert verified
So, the vector \(\overrightarrow{\mathbf{r}}\) is \(11\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 7\hat{\mathbf{k}}\). The angle between the vector \(\overrightarrow{\mathbf{r}}\) and the \(+z\) axis is given by \(acos(-7 /sqrt{(11^{2} + 5^{2} + (-7)^{2}})) degrees\), and the angle between the vectors \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) is given by \(acos ( (5*-2 + 4*2 - 6*3) / (sqrt{(5^{2} + 4^{2} + (-6)^{2}}) * sqrt{((-2)^{2} + 2^{2} + 3^{2}})) degrees\).

Step by step solution

01

Calculation of vector \(\overrightarrow{\mathbf{r}}\)

Our first task is to calculate the vector \(\overrightarrow{\mathbf{r}}\), which is defined as \(\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}\). We simply subtract the \(i\), \(j\), and \(k\) components of \(\overrightarrow{\mathbf{b}}\) from those of \(\overrightarrow{\mathbf{a}}\), and then add the corresponding components of \(\overrightarrow{\mathbf{c}}\) to the results. Thus, \(\overrightarrow{\mathbf{r}} = (5-(-2)+4)\hat{\mathbf{i}} + (4-2+3)\hat{\mathbf{j}} + (-6-3+2)\hat{\mathbf{k}} = 11\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 7\hat{\mathbf{k}}\).
02

Calculation of the angle between \(\overrightarrow{\mathbf{r}}\) and the \(+z\) axis

The angle between a vector and the \(+z\) axis can be calculated using the dot product formula. The dot product of two vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) is given by \(\overrightarrow{A} . \overrightarrow{B} = |A||B| cos(\theta)\), where \(\theta\) is the angle between the vectors. The \(+z\) axis is represented by the unit vector \(\hat{\mathbf{k}}\). Hence, \(\theta = acos (\overrightarrow{\mathbf{r}} . \hat{\mathbf{k}} / |r|\) = acos ( -7 /sqrt{(11^{2} + 5^{2} + (-7)^{2}}))\).
03

Calculation of the angle between \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\)

This is similar to step 2 and requires the use of the dot product formula. Hence, \(\theta_{ab} = acos (\overrightarrow{\mathbf{a}} . \overrightarrow{\mathbf{b}} / |a||b|\) = acos ( (5*-2 + 4*2 - 6*3) / (sqrt{(5^{2} + 4^{2} + (-6)^{2}} * sqrt{((-2)^{2} + 2^{2} + 3^{2})}))\)

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Most popular questions from this chapter

Calculate the kinetic energies of the following objects moving at the given speeds: ( \(a\) ) a \(110-\mathrm{kg}\) football linebacker running at \(8.1 \mathrm{~m} / \mathrm{s} ;\) (b) a \(4.2-\mathrm{g}\) bullet at \(950 \mathrm{~m} / \mathrm{s} ;\) ( \(c\) ) the aircraft carrier Nimitz, 91,400 tons at \(32.0\) knots.

How much power, in horsepower, must be developed by the engine of a \(1600-\mathrm{kg}\) car moving at \(26 \mathrm{~m} / \mathrm{s}(=94 \mathrm{~km} / \mathrm{h})\) on a level road if the forces of resistance total \(720 \mathrm{~N}\) ?

A spring has a force constant of \(15.0 \mathrm{~N} / \mathrm{cm} .(a)\) How much work is required to extend the spring \(7.60 \mathrm{~mm}\) from its relaxed position? ( \(b\) ) How much work is needed to extend the spring an additional \(7.60 \mathrm{~mm}\) ?

A \(106-\mathrm{kg}\) object is initially moving in a straight line with a speed of \(51.3 \mathrm{~m} / \mathrm{s}\). (a) If it is brought to a stop with a deceleration of \(1.97 \mathrm{~m} / \mathrm{s}^{2}\), what force is required, what distance does the object travel, and how much work is done by the force? \((b)\) Answer the same questions if the object's deceleration is \(4.82 \mathrm{~m} / \mathrm{s}^{2}\).

A swimmer moves through the water at a speed of \(0.22 \mathrm{~m} / \mathrm{s}\). The drag force opposing this motion is \(110 \mathrm{~N}\). How much power is developed by the swimmer?
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