Example

Direction of Motion

In a Cartesian coordinate system where $$ \hat{i} $$ denotes geographic east, $$ \hat{j} $$ denotes geographic north, and $$ \hat{k} $$ denotes altitude above sea level, a military convoy advances its position through unknown territory with velocity $$ \overset{\to }{v}=(4.0\hat{i}+3.0\hat{j}+0.1\hat{k})\text{km}\text{/}\text{h}$$. If the convoy had to retreat, in what geographic direction would it be moving?

Solution

Example

Analytical Computation of a Resultant

Three displacement vectors $$ \overset{\to }{A}$$, $$ \overset{\to }{B}$$, and $$ \overset{\to }{C} $$ in a plane ((Figure)) are specified by their magnitudes A = 10.0, B = 7.0, and C = 8.0, respectively, and by their respective direction angles with the horizontal direction $$ \alpha =35\text{°}, $$ $$\beta =-110\text{°}$$, and $$ \gamma =30\text{°}$$. The physical units of the magnitudes are centimeters. Resolve the vectors to their scalar components and find the following vector sums: (a) $$ \overset{\to }{R}=\overset{\to }{A}+\overset{\to }{B}+\overset{\to }{C}$$, (b) $$ \overset{\to }{D}=\overset{\to }{A}-\overset{\to }{B}$$, and (c) $$ \overset{\to }{S}=\overset{\to }{A}-3\overset{\to }{B}+\overset{\to }{C}$$.

Strategy

First, we use (Figure) to find the scalar components of each vector and then we express each vector in its vector component form given by (Figure). Then, we use analytical methods of vector algebra to find the resultants.

Solution

Significance

Having found the vector components, we can illustrate the vectors by graphing or we can compute magnitudes and direction angles, as shown in (Figure). Results for the magnitudes in (b) and (c) can be compared with results for the same problems obtained with the graphical method, shown in (Figure) and (Figure). Notice that the analytical method produces exact results and its accuracy is not limited by the resolution of a ruler or a protractor, as it was with the graphical method used in (Figure) for finding this same resultant.

Figure 2.24 Graphical illustration of the solutions obtained analytically in (Figure).

Example

The Tug-of-War Game

Four dogs named Ang, Bing, Chang, and Dong play a tug-of-war game with a toy ((Figure)). Ang pulls on the toy in direction $$ \alpha =55\text{°} $$ south of east, Bing pulls in direction $$ \beta =60\text{°} $$ east of north, and Chang pulls in direction $$ \gamma =55\text{°} $$ west of north. Ang pulls strongly with 160.0 units of force (N), which we abbreviate as A = 160.0 N. Bing pulls even stronger than Ang with a force of magnitude B = 200.0 N, and Chang pulls with a force of magnitude C = 140.0 N. When Dong pulls on the toy in such a way that his force balances out the resultant of the other three forces, the toy does not move in any direction. With how big a force and in what direction must Dong pull on the toy for this to happen?

Figure 2.25 Four dogs play a tug-of-war game with a toy.

Strategy

We assume that east is the direction of the positive x-axis and north is the direction of the positive y-axis. As in (Figure), we have to resolve the three given forces— $$ \overset{\to }{A} $$ (the pull from Ang), $$ \overset{\to }{B} $$ (the pull from Bing), and $$ \overset{\to }{C} $$ (the pull from Chang)—into their scalar components and then find the scalar components of the resultant vector $$ \overset{\to }{R}=\overset{\to }{A}+\overset{\to }{B}+\overset{\to }{C}$$. When the pulling force $$ \overset{\to }{D} $$ from Dong balances out this resultant, the sum of $$ \overset{\to }{D} $$ and $$ \overset{\to }{R} $$ must give the null vector $$ \overset{\to }{D}+\overset{\to }{R}=\overset{\to }{0}$$. This means that $$ \overset{\to }{D}=\text{−}\overset{\to }{R}$$, so the pull from Dong must be antiparallel to $$ \overset{\to }{R}$$.

Solution

Example

Vector Algebra

Find the magnitude of the vector $$ \overset{\to }{C} $$ that satisfies the equation $$ 2\overset{\to }{A}-6\overset{\to }{B}+3\overset{\to }{C}=2\hat{j}$$, where $$ \overset{\to }{A}=\hat{i}-2\hat{k} $$ and $$ \overset{\to }{B}=\text{−}\hat{j}+\hat{k}\text{/}2$$.

Strategy

We first solve the given equation for the unknown vector $$ \overset{\to }{C}$$. Then we substitute $$ \overset{\to }{A} $$ and $$ \overset{\to }{B}$$; group the terms along each of the three directions $$ \hat{i}$$, $$ \hat{j}$$, and $$ \hat{k}$$; and identify the scalar components $$ {C}_{x}$$, $$ {C}_{y}$$, and $$ {C}_{z}$$. Finally, we substitute into (Figure) to find magnitude C.

Solution

Example

Displacement of a Skier

Starting at a ski lodge, a cross-country skier goes 5.0 km north, then 3.0 km west, and finally 4.0 km southwest before taking a rest. Find his total displacement vector relative to the lodge when he is at the rest point. How far and in what direction must he ski from the rest point to return directly to the lodge?

Strategy

We assume a rectangular coordinate system with the origin at the ski lodge and with the unit vector $$ \hat{i} $$ pointing east and the unit vector $$ \hat{j} $$ pointing north. There are three displacements: $$ {\overset{\to }{D}}_{1}$$, $$ {\overset{\to }{D}}_{2}$$, and $$ {\overset{\to }{D}}_{3}$$. We identify their magnitudes as $$ {D}_{1}=5.0\,\text{km}$$, $$ {D}_{2}=3.0\,\text{km}$$, and $$ {D}_{3}=4.0\,\text{km}$$. We identify their directions are the angles $$ {\theta }_{1}=90\text{°}$$, $$ {\theta }_{2}=180\text{°}$$, and $$ {\theta }_{3}=180\text{°}+45\text{°}=225\text{°}$$. We resolve each displacement vector to its scalar components and substitute the components into (Figure) to obtain the scalar components of the resultant displacement $$ \overset{\to }{D} $$ from the lodge to the rest point. On the way back from the rest point to the lodge, the displacement is $$ \overset{\to }{B}=\text{−}\overset{\to }{D}$$. Finally, we find the magnitude and direction of $$ \overset{\to }{B}$$.

Solution

Significance

Notice that no figure is needed to solve this problem by the analytical method. Figures are required when using a graphical method; however, we can check if our solution makes sense by sketching it, which is a useful final step in solving any vector problem.

Example

Displacement of a Jogger

A jogger runs up a flight of 200 identical steps to the top of a hill and then runs along the top of the hill 50.0 m before he stops at a drinking fountain ((Figure)). His displacement vector from point A at the bottom of the steps to point B at the fountain is $$ {\overset{\to }{D}}_{AB}=(-90.0\hat{i}+30.0\hat{j})\text{m}$$. What is the height and width of each step in the flight? What is the actual distance the jogger covers? If he makes a loop and returns to point A, what is his net displacement vector?

Figure 2.26 A jogger runs up a flight of steps.

Strategy

The displacement vector $$ {\overset{\to }{D}}_{AB} $$ is the vector sum of the jogger’s displacement vector $$ {\overset{\to }{D}}_{AT} $$ along the stairs (from point A at the bottom of the stairs to point T at the top of the stairs) and his displacement vector $$ {\overset{\to }{D}}_{TB} $$ on the top of the hill (from point T at the top of the stairs to the fountain at point B). We must find the horizontal and the vertical components of $$ {\overset{\to }{D}}_{TB}$$. If each step has width w and height h, the horizontal component of $$ {\overset{\to }{D}}_{TB} $$ must have a length of 200w and the vertical component must have a length of 200h. The actual distance the jogger covers is the sum of the distance he runs up the stairs and the distance of 50.0 m that he runs along the top of the hill.

Solution

((Figure)).

Example

The Unit Vector of Direction

If the velocity vector of the military convoy in (Figure) is $$ \overset{\to }{v}=(4.000\hat{i}+3.000\hat{j}+0.100\hat{k})\text{km}\text{/}\text{h}$$, what is the unit vector of its direction of motion?

Strategy

The unit vector of the convoy’s direction of motion is the unit vector $$ \hat{v} $$ that is parallel to the velocity vector. The unit vector is obtained by dividing a vector by its magnitude, in accordance with (Figure).

Solution

Significance

Note that when using the analytical method with a calculator, it is advisable to carry out your calculations to at least three decimal places and then round off the final answer to the required number of significant figures, which is the way we performed calculations in this example. If you round off your partial answer too early, you risk your final answer having a huge numerical error, and it may be far off from the exact answer or from a value measured in an experiment.