Verified answer

You should INSTANTLY recognize a 3,4,5 triangle. So the other side has a magnitude of 7.5 km

But if you don't then you can resort to Pythagoras' theorem which is the standard method for all vector synthesis.

c = sqrt( a^2 + b^2 ) = sqrt( 4.5^2 + 6^2) = ( insert drum roll here ) 7.5 km

Now you can see that the angle will be south of west.

to get the angle you use tan(theta) = opposite /adjacent

theta = atan( 6/4.5) = atan (4/3) = 53 degrees South of West

Since south and west are perpendicular to each other, we can use the Pythagorean theorem to determine the magnitude of this velocity.

Speed = √(4.50^2 + 6.00^2) = 7.5 km

Tan θ = 6 ÷ 4.5 = 1⅓

The angle is approximately 53˚ south of west or 37 west of south. This is 3,4,5 right triangle.

using a vector calculator and compass directions

[4.50 km , 270°] + [6.00 km , 180°] = [ 7.5 km , 216.9°]

or

7.5 km at 36.9° west of south

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