# 2.4 Products of vectors  (Page 8/16)

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Find the angles that vector $\stackrel{\to }{D}=\left(2.0\stackrel{^}{i}-4.0\stackrel{^}{j}+\stackrel{^}{k}\right)\text{m}$ makes with the x -, y -, and z - axes.

${\theta }_{i}=64.12\text{°},{\theta }_{j}=150.79\text{°},{\theta }_{k}=77.39\text{°}$

Show that the force vector $\stackrel{\to }{D}=\left(2.0\stackrel{^}{i}-4.0\stackrel{^}{j}+\stackrel{^}{k}\right)\text{N}$ is orthogonal to the force vector $\stackrel{\to }{G}=\left(3.0\stackrel{^}{i}+4.0\stackrel{^}{j}+10.0\stackrel{^}{k}\right)\text{N}$ .

Assuming the + x -axis is horizontal to the right for the vectors in the previous figure, find the following vector products: (a) $\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{C}$ , (b) $\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{F}$ , (c) $\stackrel{\to }{D}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{C}$ , (d) $\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\stackrel{\to }{F}+2\stackrel{\to }{C}\right)$ , (e) $\stackrel{^}{i}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ , (f) $\stackrel{^}{j}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ , (g) $\left(3\stackrel{^}{i}-\stackrel{^}{j}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ , and (h) $\stackrel{^}{B}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ .

a. $-119.98\stackrel{^}{k}$ , b. $-173.2\stackrel{^}{k}$ , c. $+93.69\stackrel{^}{k}$ , d. $-413.2\stackrel{^}{k}$ , e. $+39.93\stackrel{^}{k}$ , f. $-30.09\stackrel{^}{k}$ , g. $+149.9\stackrel{^}{k}$ , h. 0

Find the cross product $\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{C}$ for (a) $\stackrel{\to }{A}=2.0\stackrel{^}{i}-4.0\stackrel{^}{j}+\stackrel{^}{k}$ and $\stackrel{\to }{C}=3.0\stackrel{^}{i}+4.0\stackrel{^}{j}+10.0\stackrel{^}{k}$ , (b) $\stackrel{\to }{A}=3.0\stackrel{^}{i}+4.0\stackrel{^}{j}+10.0\stackrel{^}{k}$ and $\stackrel{\to }{C}=2.0\stackrel{^}{i}-4.0\stackrel{^}{j}+\stackrel{^}{k}$ , (c) $\stackrel{\to }{A}=-3.0\stackrel{^}{i}-4.0\stackrel{^}{j}$ and $\stackrel{\to }{C}=-3.0\stackrel{^}{i}+4.0\stackrel{^}{j}$ , and (d) $\stackrel{\to }{C}=-2.0\stackrel{^}{i}+3.0\stackrel{^}{j}+2.0\stackrel{^}{k}$ and $\stackrel{\to }{A}=-9.0\stackrel{^}{j}$ .

For the vectors in the earlier figure, find (a) $\left(\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{F}\right)·\stackrel{\to }{D}$ , (b) $\left(\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{F}\right)·\left(\stackrel{\to }{D}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}\right)$ , and (c) $\left(\stackrel{\to }{A}·\stackrel{\to }{F}\right)\left(\stackrel{\to }{D}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}\right)$ .

a. 0, b. 173,194, c. $+199,993\stackrel{^}{k}$

(a) If $\stackrel{\to }{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{F}=\stackrel{\to }{B}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{F}$ , can we conclude $\stackrel{\to }{A}=\stackrel{\to }{B}$ ? (b) If $\stackrel{\to }{A}·\stackrel{\to }{F}=\stackrel{\to }{B}·\stackrel{\to }{F}$ , can we conclude $\stackrel{\to }{A}=\stackrel{\to }{B}$ ? (c) If $F\stackrel{\to }{A}=\stackrel{\to }{B}F$ , can we conclude $\stackrel{\to }{A}=\stackrel{\to }{B}$ ? Why or why not?

You fly $32.0\phantom{\rule{0.2em}{0ex}}\text{km}$ in a straight line in still air in the direction $35.0\text{°}$ south of west. (a) Find the distances you would have to fly due south and then due west to arrive at the same point. (b) Find the distances you would have to fly first in a direction $45.0\text{°}$ south of west and then in a direction $45.0\text{°}$ west of north. Note these are the components of the displacement along a different set of axes—namely, the one rotated by $45\text{°}$ with respect to the axes in (a).

a. 18.4 km and 26.2 km, b. 31.5 km and 5.56 km

Rectangular coordinates of a point are given by (2, y ) and its polar coordinates are given by $\left(r,\pi \text{/}6\right)$ . Find y and r .

If the polar coordinates of a point are $\left(r,\phi \right)$ and its rectangular coordinates are $\left(x,y\right)$ , determine the polar coordinates of the following points: (a) (− x , y ), (b) (−2 x , −2 y ), and (c) (3 x , −3 y ).

a. $\left(r,\phi +\pi \text{/}2\right)$ , b. $\left(2r,\phi +2\pi \right)$ , (c) $\left(3r,\text{−}\phi \right)$

Vectors $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ have identical magnitudes of 5.0 units. Find the angle between them if $\stackrel{\to }{A}+\stackrel{\to }{B}=5\sqrt{2}\stackrel{^}{j}$ .

Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4.76 nautical miles (nmi) in a direction $37\text{°}$ north of east to Noi. From Noi, it sails $69\text{°}$ west of north to Poi. On its return leg from Poi, it sails $28\text{°}$ east of south. What distance does the boat sail between Noi and Poi? What distance does it sail between Moi and Poi? Express your answer both in nautical miles and in kilometers. Note: 1 nmi = 1852 m.

${d}_{\text{PM}}=33.12\phantom{\rule{0.2em}{0ex}}\text{nmi}=61.34\phantom{\rule{0.2em}{0ex}}\text{km},\phantom{\rule{0.4em}{0ex}}{d}_{\text{NP}}=35.47\phantom{\rule{0.2em}{0ex}}\text{nmi}=65.69\phantom{\rule{0.2em}{0ex}}\text{km}$

An air traffic controller notices two signals from two planes on the radar monitor. One plane is at altitude 800 m and in a 19.2-km horizontal distance to the tower in a direction $25\text{°}$ south of west. The second plane is at altitude 1100 m and its horizontal distance is 17.6 km and $20\text{°}$ south of west. What is the distance between these planes?

Show that when $\stackrel{\to }{A}+\stackrel{\to }{B}=\stackrel{\to }{C}$ , then ${C}^{2}={A}^{2}+{B}^{2}+2AB\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\phi$ , where $\phi$ is the angle between vectors $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ .

proof

Four force vectors each have the same magnitude f . What is the largest magnitude the resultant force vector may have when these forces are added? What is the smallest magnitude of the resultant? Make a graph of both situations.

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