Option 23 DHS 2.1

DHS - 2.1
No. 1.23. 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 = 5; β = 4; γ = -6; δ = 2; k = 2; ℓ = 9; φ = 2π / 3; λ = 3; μ = 2; ν = 1; τ = -1/2.
No. 2.23. 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 (3; 4; 1); B (5; –2; 6); C (4; 2; –7); .......
No. 3.23. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (1; 2; 3); b (–5; 3; –1); c (–6; 4; 5); d (–4; 11; 20)