RAP IDZ 2.1 option 9

IDZ - 2.1
No. 1.9. The vectors are given by 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 = -2; γ = 1; δ = 5; k = 3; ℓ = 6; φ = 4π / 3; λ = -1; μ = 2; ν = 1; τ = 1.
No. 2.9. From the coordinates of the points A; B and C for these vectors, find: a) the modulus of the vector a; b) scalar product of vectors a and b; c) the projection of the vector c onto the vector d; d) the coordinates of the point M of the dividing segment ℓ with respect to α:.
Given: A (3; 4; -4); B (-2; 1; 2); C (2; -3; 1); .......
No. 3.9. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (0; 2; -3); b (4; -3; -2); c (-5; -4; 0); d (-19; -5; -4).