Option 12 DHS 2.1

DHS - 2.1
№ 1.12. 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 = -2; β = -4; γ = 3; δ = 6; k = 3; ℓ = 2; φ = 7π / 3; λ = -1/2; μ = 3; ν = 1; τ = 2.
№ 2.12. 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 point M; dividing the segment ℓ in relation to α :.
Given: A (-2; -3; -2); B (1; 4; 2); C (1; –3; 3); .......
№ 3.12. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (3; 1; -3); b (–2; 4; 1); c (1; –2; 5); d (1; 12; -20).