Let u = ⟨–2, 6⟩, v = ⟨3, 2⟩, and w = ⟨–1, 4⟩. What is the magnitude and direction angle of 2(u + v) – 3w? 2.8; θ = 45.0°
6.4; θ = 38.7°
28.0; θ = 92.0°
41.0; θ = 51.3°

u = ⟨-2, 6⟩

v = ⟨3, 2⟩

w = ⟨-1, 4⟩

u + v = ⟨-2, 6⟩ + ⟨3, 2⟩

u + v = ⟨-2 + 3, 6 + 2⟩

u + v = ⟨1, 8⟩

2 (u + v) = 2 ⟨-1, 8⟩

2 (u + v) = ⟨2 • 1, 2 • 8⟩

2 (u + v) = ⟨2, 16⟩

-3 w = -3 ⟨-1, 4⟩

-3 w = ⟨-3 • (-1), -3 • 4⟩

-3 w = ⟨3, -12⟩

2 (u + v) - 3 w = 2 (u + v) + (-3 w)

2 (u + v) - 3 w = ⟨2, 16⟩ + ⟨3, -12⟩

2 (u + v) - 3 w = ⟨2 + 3, 16 + (-12)⟩

2 (u + v) - 3 w = ⟨5, 4⟩

Then the magnitude of this vector is

||2 (u + v) - 3 w|| = √(5² + 4²) = √41 ≈ 6.403

so the second choice is correct.

a

step-by-step explanation: