Option 27 DHS 2.1

DHS - 2.1
No. 1.27. Given the vector a = α · m + β · n; b = γ · m + δ · n; | m | = k; | n | = ℓ; (m; n) = φ;
Find: a) (λ · a + μ · b) · (ν · a + τ · b); b) the projection (ν · a + τ · b) on b; c) cos (a + τ · b).
Given: α = -3; β = 4; γ = 5; δ = -6; k = 4; ℓ = 5; φ = π; λ = 2; μ = 3; ν = -3; τ = -1.
No. 2.27. The coordinates of points A; B and C for the indicated vectors to find: a) the modulus of the vector a; b) the scalar product of vectors a and b; c) the projection of the vector c on the vector d; d) coordinates of the point M; dividing the segment ℓ in relation to α :.
Given: A (6; 5; –4); B (–5; –2; 2); C (3; 3; 2); .......
No. 3.27. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (4; 5; 1); b (1; 3; 1); c (–3; –6; 7); d (19; 33; 0)