We are given two vectors in magnitude , angle form , and want to find the component form of them.
We recall that the x-component of a vector is given by the cosine projection of its magnitude, while the y-component is given by the sine projection:
Then in general terms:
x-component = |V| cos (angle)
y-component = |V| sin(angle)
where |V| representes the magnitude of the given vector.
So for our cases:
Case a)
the magnitude is 16 and the angle is 225 degrees, therefore:
x-component = 16 * cos(225) = -11.3137...
y-component - 16 * sin(225) = -11.3137....
this is telling us that the vector is on the third quadrant and corresponds to equal x and y components. So we could have expressed them in exact form using the fact that the sin and cos of that angle can be expressed by square root of 2 divided by 2.
So if they want the EXACT value, you write : -16 * root(2) / 2 = - 8 root(2)for each one.
Case b):
12 is the magnitude and 63 degrees the angle. Then:
x-component = 12 * cos(63) = 5.4479
y-component = 12 * sin(63) = 10.6921
So this vector is on the first quadrant, and the angloe is not a special angle so we don't have an easy square root expression for the sine and cosine.