RAP IDZ 2.1 option 5

IDZ - 2.1
№ 1.5. 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 = -4; δ = 5; k = 2; ℓ = 3; φ = π / 3; λ = 2; μ = -3; ν = 4; τ = 1.
No. 2.5. 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) coordinates
points M of the dividing segment ℓ with respect to α :.
Given: A (2; 4; 5); In (1; -2; 3); C (-1; -2; 4); .......
No. 3.5. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (1; -1; 1); b (-5; -3; 1); c (2; -1; 0); d (-15; -10; 5).