Option 20 DHS 2.1

DHS - 2.1
No. 1.20. 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 = -5; γ = -2; δ = 3; k = 1; ℓ = 6; φ = 3π / 2; λ = 4; μ = 5; ν = 1; τ = -2.
No. 2.20. 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 (5; 4; 4); B (–5; 2; 3); C (4; 2; –5); .......
No. 3.20. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (11; 1; 2); b (–3; 3; 4); c (–4; –2; 7); d (–5; 11; –15)