Option 19 DHS 2.1

DHS - 2.1
No. 1.19. 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 = 4; β = -5; γ = -1; δ = 3; k = 6; ℓ = 3; φ = 2π / 3; λ = 2; μ = -5; ν = 1; τ = 2.
No. 2.19. 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
points M; dividing the segment ℓ in relation to α :.
Given: A (–4; –2; –5); B (3; 7; 2); C (4; 6; –3)
No. 3.19. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (5; 3; 2); b (2; –5; 1); c (–7; 4; –3); d (36; 1; 15)