Question

An object experiences two velocity vectors in its environment.

v1 = −55i + 3j
v2 = 4i + 13j

What is the true speed and direction of the object? Round the speed to the thousandths place and the direction to the nearest degree.

53.451; 163°
53.451; 17°
48.425; 163°
48.425; 17°

Answer 1

53.451; 163°

Explanation:

Answer 2

(a) 53.451∠163°

Explanation:

The sum of the vectors is the sum of their components:

v1 +v2 = (-55i +3j) +(4i +13j) = (-55 +4)i +(3 +13)j = -51i +16j

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conversion to polar form

This is converted to magnitude and direction by ...

|ai +bj| = √(a² +b²)

|-51i +16j| = √((-51)² +16²) = √(2601 +256) = √2857 ≈ 53.451

and

∠(ai +bj) = arctan(b/a) = arctan(16/-51) ≈ -17° +180° = 163°

note that 180° is added because the components indicate the angle is in the 2nd quadrant.

The object's speed and direction are ...

53.451 ∠163°

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Additional comment

If you're doing a lot of vector calculations, the process is simplified immensely by a suitable calculator. (See attached.) 2-dimensional vectors are handled nicely using the complex number functions of the calculator.