(C) 6.08∠175.28°
Step-by-step explanation:
There are a couple of ways to find the sum of the vectors. One is to add their components, then convert back to polar form. Another is to make use of the law of cosines.
Adding components
The components of a vector with length z at some angle "a" are ...
z(cos(a), sin(a)) = (z·cos(a), z·sin(a))
For the given vectors, the rectangular components are ...
(7·cos(150°), 7·sin(150°)) ≈ (-6.062178, 3.5)
(3·cos(-90°), 3·sin(-90°)) = (0, -3)
The sum of these is ...
(-6.062178, 3.5) +(0, -3) = (-6.062178, 0.5)
If you recognize that these are the coordinates of a point just above the -x axis at a distance of only slightly more than 6.06 from the origin, you can correctly choose choice C as the appropriate answer. Or, you can compute the magnitude and angle.
The magnitude of this vector is the distance of its end point from the origin:
magnitude = √((-6.062178)² + 0.5²) = √37 ≈ 6.08276
The angle is found using the arctangent function. We recognize this will be a 2nd quadrant angle.
angle = arctan(0.5/-6.062178) ≈ 175.28499°
So, the vector sum is ...
6.08∠175.28° . . . . matches choice C
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Law of Cosines
Moving the vector of length 3 to the end of the one of length 7, we see that the angle between those vectors is 60°. Then we can use the law of cosines to find the magnitude of the resultant:
r² = 3² +7² -2·3·7·cos(60°) = 37
r = √37 ≈ 6.08 . . . . matches choice C
The angle can be found from the law of sines. The angle (A) between the resultant and the vector of length 7 is ...
sin(A)/3 = sin(60°)/6.08
sin(A) = 3/6.08·sin(60°) ≈ .427121
A = arcsin(.427121) ≈ 25.28°
Then the direction of the resultant is this angle added to the 150° angle of the vector of length 7. The resultant direction is ...
angle = A+150° = 175.28°
The resultant is about 6.08∠175.28°. (matches choice C)
Comment on the answer
Our calculation gives the angle as 175.28499604605178°. This value rounds to 175.28°. If you round to 3, 4, or 5 digits, you will see 175.28500 and might conclude that the correct answer is 175.29°. This will be the case for a calculator that only displays 8 significant digits. Using a calculator that displays 10 or more digits, you will see that the correctly rounded answer is 175.28°.
