## Adding Vectors Using The Component Method

by

Betty Roombos

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A vector of the form 6 units 300 N of E may be broken down into its north and east components and then be combined with other components of the same type.  Since 6 units is the hypotenuse of a right triangle, the north and east components are the legs of this right triangle.  The first direction listed is always found using the hypotenuse and the sine of the angle and the second direction listed is always found using the hypotenuse and the cosine of the angle.

With the above information, let's add the following displacement vectors to find the resultant:  4 km 250 E of S and 7 km 350 W of N.

The east component of the first displacement is 4(sin 250) = 1.69 km E

The south component of the first displacement is 4(cos 250) = 3.63 km S

The west component of the second displacement is 7(sin 350) = 4.02 km W

The north component of the second displacement is 7(cos 350) = 5.73 km N

Now the components in the same or opposite directions may be combined.  To be consistent let's make east and north the positive directions.  Then 1.69 km E - 4.02 km W = -2.33 and since the answer is negative, this means that the direction is west or 2.33 km W.   Similarly, -3.63 km S + 5.73 km N = +2.10 km, and since the answer is positive, the direction is north or 2.10 km N.

2.33 km W and 2.10 km N are now the legs of a right triangle, and the hypotenuse of this right triangle is the sum of the original displacements.  Therefore, we may use the Pythagorean theorem to find the length of the hypotenuse and we may use tangent to find the angle.

R = [ (2.33 km)2 + (2.10 km)2]1/2 = 3.14 km

Then the tangent of the angle = either 2.33 km W / 2.10 km N  or 2.10 km N / 2.33 km W and the corresponding angles are 480 and 42.00 respectively.

The final answer is then either 3.14 km 480 W of N or 3.14 km 420 N or W.  Notice that the direction listed in the numerator is written first and that the direction in the denominator is listed second.

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