Option 14 DHS 2.1

Option 14 DHS 2.1

DHS - 2.1
No. 1.14. 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 = 3; γ = 5; δ = 1; k = 2; ℓ = 5; φ = 2π; λ = -3; μ = 4; ν = 2; τ = 3.
No. 2.14. 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 (10; 6; 3); B (-2; 3; 5); C (3; –4; -6); .......
No. 3.14. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (4; 2; 3); b (–3; 1; –8); c (2; –4; 5); d (-12; 14; -31).