How do you find the resultant vector?

Question

Two tugboats are needed to bring a tanker into harbor. Tugboat A heads at 120° going 12 mph, and Tugboat B goes 15 mph with a direction of 72°. Find the resultant vector. At what speed is the tanker being pulled, and in what direction?

Answer 1

Tugboat A is going 12 (cos120º, sin120º)

Tugboat B is going 15(cos72º, sin 72º)

 

Resultant vector is 12(cos120º, sin120º) +  15(cos72º, sin 72º)

= (12cos120º + 15cos72º, 12sin120º + 15sin72º)

= (-6 + 4.6352, 10.3923 + 14.2658)

= (-1.3648, 24.6581)

 

magnitude (speed) = √(-1.3648^2 + 24.6581^2)

= √(1.8627  + 608.0219)

= √609.8846

= 24.70

 

direction is found using the Arctan function, but before you apply it, note that the boat is headed in a direction in the II Quadrant, because the Arctan function only gives you answers in the I and IV Quadrants, so you will need to adjust the Arctan result for that fact.

 

Arctan(24.6581/-1.3648) = Artan (-18.07) = -86.83º  which is a IV Quadrant answer.  

 

The corresponding answer in the II Quadrant is -86.83º + 180º = 93.17º