$$\begin{gathered} \mathbf{If}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{c}\right)\mathbf{}\mathbf{,}\mathbf{}\mathbf{does}\mathbf{}\left(a\right)\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{d}\right)\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{b}\right)\mathbf{}\mathbf{,}\mathbf{}\left(b\right)\mathbf{}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{=}\left(-\stackrel{\rightharpoonup }{b}\right)\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}} \\ \stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{d}\right)\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{?} \end{gathered}$$

A ball is to be shot from level ground with a certain speed. Figure 4-45 shows the range R it will have versus the launch angle${\mathit{\theta }}_{\mathbf{0}}$. The value of${\mathit{\theta }}_{\mathbf{0}}$ determines the flight time; let${\mathit{t}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ represent the maximum flight time. What is the least speed the ball will have during its flight if${\mathit{\theta }}_{\mathbf{0}}$ is chosen such that flight time is$\mathbf{0}\mathbf{.}\mathbf{500}{\mathit{t}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$?

In Fig. , a cube of edge length a sits with one corner at the origin of an xyz coordinate system. A body diagonal is a line that extends from one corner to another through the center. In unit-vector notation, what is the body diagonal that extends from the corner at (a) coordinates (0,0,0), (b) coordinates (a,0,0), (c) coordinates (0,a,0), and (d) coordinates (a,a,0)? (e) Determine the angles that the body diagonals make with the adjacent edges. (f) Determine the length of the body diagonals in terms of a.

$$\begin{gathered} \mathbf{Three}\mathbf{}\mathbf{vectors}\mathbf{}\mathbf{are}\mathbf{}\mathbf{given}\mathbf{}\mathbf{by}\mathbf{}\mathbf{a}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\overrightarrow{\mathbf{a}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{}\overrightarrow{\mathbf{a}} \\ \mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{,}\mathbf{}\overrightarrow{\mathbf{c}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}\widehat{\mathbf{k}\mathbf{.}} \\ \mathbf{Find}\mathbf{}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\overrightarrow{\mathbf{a}}\mathbf{.}\mathbf{(}\overrightarrow{\mathbf{a}}\mathbf{\times }\overrightarrow{\mathbf{c}}\mathbf{)}\mathbf{,}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\overrightarrow{\mathbf{a}}\mathbf{.}\mathbf{(}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{)}\mathbf{}\mathbf{,}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\overrightarrow{\mathbf{a}}\mathbf{\times }\mathbf{(}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{)}\mathbf{.} \end{gathered}$$

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 $\overrightarrow{\mathbf{A}\mathbf{B}}$is along the plane of the fault.The horizontal component of is the strike-slip AC. The component $\overrightarrow{\mathbf{A}\mathbf{B}}$of that is directed down the plane of the fault is the dip-slip AD.(a) What is the magnitude of the net displacement$\overrightarrow{\mathbf{A}\mathbf{B}}$if the strike-slip is 22.0 m and the dip-slip is 17.0 m ? (b) If the plane of the fault is inclined at angle$\mathit{\varphi }\mathbf{=}\mathbf{52}\mathbf{.}\mathbf{0}\mathbf{°}$to the horizontal, what is the vertical component of$\overrightarrow{\mathbf{A}\mathbf{B}}$?

For the following three vectors, what is $\mathbf{3}\overrightarrow{\mathbf{C}}\mathbf{.}\left(2\overrightarrow{A}\times \overrightarrow{B}\right)$?

$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{00}\hat{\mathbf{k}}\mathbf{,}\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{k}}\mathbf{,}\overrightarrow{\mathbf{C}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}$