# 3.2 Vector addition and subtraction: graphical methods  (Page 6/16)

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Explain why it is not possible to add a scalar to a vector.

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

## Problems&Exercises

Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.

Find the following for path A in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

(a) $\text{480 m}$

(b) $\text{379 m}$ , $\text{18.4°}$ east of north

Find the following for path B in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

Find the north and east components of the displacement for the hikers shown in [link] .

north component 3.21 km, east component 3.83 km

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ , as in [link] , then this problem asks you to find their sum .)

Suppose you first walk 12.0 m in a direction $\text{20°}$ west of north and then 20.0 m in a direction $\text{40.0°}$ south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements $\mathbf{A}$ and $\mathbf{B}$ , as in [link] , then this problem finds their sum .)

$\text{19}\text{.}\text{5 m}$ , $4\text{.}\text{65°}$ south of west

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg $\mathbf{B}$ , which is 20.0 m in a direction exactly $\text{40°}$ south of west, and then leg $\mathbf{A}$ , which is 12.0 m in a direction exactly $\text{20°}$ west of north. (This problem shows that $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$ .)

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction $\text{40.0°}$ north of east (which is equivalent to subtracting $\mathbf{\text{B}}$ from $\mathbf{A}$ —that is, to finding $\mathbf{\text{R}}\prime =\mathbf{\text{A}}-\mathbf{\text{B}}$ ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction $\text{40.0°}$ south of west and then 12.0 m in a direction $\text{20.0°}$ east of south (which is equivalent to subtracting $\mathbf{\text{A}}$ from $\mathbf{\text{B}}$ —that is, to finding $\mathbf{\text{R}}\prime \prime =\mathbf{\text{B}}-\mathbf{\text{A}}=-\mathbf{\text{R}}\prime$ ). Show that this is the case.

(a) $\text{26}\text{.}\text{6 m}$ , $\text{65}\text{.}\text{1°}$ north of east

(b) $\text{26}\text{.}\text{6 m}$ , $\text{65}\text{.}\text{1°}$ south of west

Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors $\mathbf{A}$ , $\mathbf{B}$ , and $\mathbf{C}$ , all having different lengths and directions. Find the sum then find their sum when added in a different order and show the result is the same. (There are five other orders in which $\mathbf{A}$ , $\mathbf{B}$ , and $\mathbf{C}$ can be added; choose only one.)

Show that the sum of the vectors discussed in [link] gives the result shown in [link] .

$\text{52}\text{.}\text{9 m}$ , $\text{90}\text{.}\text{1°}$ with respect to the x -axis.

Find the magnitudes of velocities ${v}_{\text{A}}$ and ${v}_{\text{B}}$ in [link]

Find the components of ${v}_{\text{tot}}$ along the x - and y -axes in [link] .

x -component 4.41 m/s

y -component 5.07 m/s

Find the components of ${v}_{\text{tot}}$ along a set of perpendicular axes rotated $\text{30°}$ counterclockwise relative to those in [link] .

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
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