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## Q. 1.4

Figure (a) shows two position vectors of magnitudes A = 60 ft and  B = 100 ft. (A position vector is a vector drawn between two points in space.) Determine the resultant R = A + B using the following methods: (1) analytically, using the triangle law; and (2) graphically, using the triangle law.

## Verified Solution

Part 1

The first step in the analytical solution is to draw a sketch (approximately to scale) of the triangle law. The magnitude and direction of the resultant are then found by applying the laws of sines and cosines to the triangle.

In this problem, the triangle law for the vector addition of A and B is shown in Fig. (b). The magnitude R of the resultant and the angle α are the unknowns to be determined. Applying the law of cosines, we obtain

$R^{2}=60^{2}+100^{2}-2(60)(100)\:cos\:140^\circ$

which yields $R = 151.0\:ft$.
The angle $α$ can now be found from the law of sines:

$\frac{100}{\sin\alpha} =\frac{R}{\sin140^\circ}$

Substituting $R = 151.0\:ft$ and solving for $α$, we get $α = 25.2^\circ$. Referring to $Fig.\:(b)$, we see that the angle that $R$ makes with the horizontal is $30^\circ+α=30^\circ+25.2^\circ=55.2^\circ$. Therefore, the resultant of $A$ and $B$ is

$R = 151.0\:ft$

Part 2

In the graphical solution, $Fig.\:(b)$ is drawn to scale with the aid of a ruler and a protractor. We first draw the vector $A$ at $30^\circ$ to the horizontal and then append vector $B$ at $70^\circ$ to the horizontal. The resultant $R$ is then obtained by drawing a line from the tail of $A$ to the head of $B$. The magnitude of $R$ and the angle it makes with the horizontal can now be measured directly from the figure.

Of course, the results would not be as accurate as those obtained in the analytical solution. If care is taken in making the drawing, two-digit accuracy is the best we can hope for. In this problem we should get $R≈150\:ft$, inclined at $55^\circ$ to the horizontal.