IDZ 2.1 - Option 14. Decisions Ryabushko AP

1. Given the vectors a = αm + βn and b = γm + δn, where | m | = k; | n | = l; (m, ^ n) = φ.
Find a) (λa + μb) ∙ (νa + τb); b) PRV (νa + τb); a) cos (a, ^ τb)
1.14 α = -2, β = 3, γ = 5, δ = 1, k = 2, l = 5, φ = 2π, λ = -3, μ = 4, ν = 2, τ = 3

2. The coordinates of points A, B and C for the indicated vectors to find: a) a unit vectors; b) the inner product of vectors a and b; c) the projection of c-vector d; z) coordinates of the point M, dividing the segment l against α: β
2.14 A (10, 6, 3), B (-2, 4, 5), C (3, -4, -6) a = 5AC - 2CB, b = c = BA, d = AC, l = CB, α = 1, β = 5

3. Prove that the vectors a, b, c form a basis, and to find the coordinates of the vector d in this basis.
3.14 a = (4, 2, 3); b = (-3, 1, -8); c = (2, -4, 5); d = (-12, 14, -31)