Question

Given the vector u with magnitude 3 and direction, 45°, and vector v with magnitude 2 and direction 350°, find the magnitude and direction of the sun u+v. write the sum u+v in magnitude and direction form with the magnitude rounded to the nearest tenth in the direction rounded to the nearest degree

Answer 1

4.5∠23°

Explanation:

You want the sum of 3∠45° and 2∠350° in magnitude and direction form.

Sum

The attached calculator image shows the sum is ...

3∠45° +2∠350° ≈ 4.5∠23°

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Additional comment

You need a calculator for the sum in any event, so it is appropriate to use the features of the calculator that are directly helpful.

Often, you are asked to do the sum using rectangular coordinates:

3∠45° = (3cos(45°), 3sin(45°)) ≈ {2.1213, 2.1213}

2∠350° = (2cos(350°), 2sin(350°)) ≈ (1.9696, -0.3473}

Then the sum is (2.1213+1.9696, 2.1213-0.3473) = (4.0909, 1.7740)

The magnitude is the Pythagorean sum of these coordinates:

magnitude = √(4.0909² +1.7740²) ≈ 4.5

The angle is the arctangent of their ratio:

angle = arctan(1.7740/4.0909) ≈ 23°

Alternatively, you can solve the triangle of the two vectors and their sum. The diagram of this is shown in the second attachment. The law of cosines and the law of sines are used.

c = √(a² +b² -2ab·cos(C)) ≈ √(3² +2² -2·3·2·cos(125°)) ≈ √19.883 ≈ 4.545

The angle of 'c' is 45° less the interior angle of the triangle at O.

45° - arcsin(2/4.459·sin(125°)) ≈ 45° -21.6° ≈ 23.4°

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