To find the results of the given vector operations, we can simply add or subtract the corresponding components of the vectors involved. Here are the calculations:
u - v + w:
(3, 2) - (4, -3) + (-2, 5)
= (3 - 4 - 2, 2 - (-3) + 5)
= (-3, 10)
2u + v - 3w:
2(3, 2) + (4, -3) - 3(-2, 5)
= (6, 4) + (4, -3) - (-6, 15)
= (6 + 4 + 6, 4 - 3 - 15)
= (16, -14)
0.5(u + 2v + w):
0.5[(3, 2) + 2(4, -3) + (-2, 5)]
= 0.5[(3, 2) + (8, -6) + (-2, 5)]
= 0.5[(3 + 8 - 2, 2 - 6 + 5)]
= 0.5[(9, 1)]
= (4.5, 0.5)
So, the results of the given vector operations are:
u - v + w = (-3, 10)
2u + v - 3w = (16, -14)
0.5(u + 2v + w) = (4.5, 0.5)