Question

Find the decomposition v = v∥ v⊥ with respect to u.1. v = ⟨4, −1, 0⟩, u = ⟨0, 1, 1⟩.2. v = ⟨x, y⟩, u = ⟨1, −1⟩.

Answer 1

To find the decomposition v = v∥ + v⊥ with respect to u, you can use the formulas: v∥ = (v · u/|u|²)u and v⊥ = v - v∥. Applying these formulas to the given vectors v and u, we can find the parallel and perpendicular components.

Step-by-step explanation:

To find the decomposition v = v∥ + v⊥ with respect to u, we need to determine the parallel component of v along u (v∥) and the perpendicular component of v to u (v⊥). To calculate v∥, we can use the formula v∥ = (v · u/|u|^2)u. To calculate v⊥, we can use the formula v⊥ = v - v∥. Let's apply these formulas to the given vectors:

1. v = ⟨4, -1, 0⟩, u = ⟨0, 1, 1⟩
   v∥ = (v · u/|u|^2)u = (4*0 + (-1)*1 + 0*1) / ((0)^2 + (1)^2 + (1)^2) ⟨0, 1, 1⟩ = ⟨0, 1/2, 1/2⟩
   v⊥ = v - v∥ = ⟨4, -1, 0⟩ - ⟨0, 1/2, 1/2⟩ = ⟨4, -3/2, -1/2⟩
2. v = ⟨x, y⟩, u = ⟨1, -1⟩
   v∥ = (v · u/|u|^2)u = (x*1 + y*(-1)) / ((1)^2 + (-1)^2) ⟨1, -1⟩ = ⟨x-y, -x+y⟩
   v⊥ = v - v∥ = ⟨x, y⟩ - ⟨x-y, -x+y⟩ = ⟨2x-y, x+2y⟩