Option 13 DHS 2.1

DHS - 2.1
No. 1.13. 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 = 3; γ = -1; δ = 2; k = 4; ℓ = 5; φ = 3π / 2; λ = 2; μ = - 3; ν = 1; τ = 2.
No. 2.13. 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 (5; 6; 1); B (-5; 2; 6); C (3; –3; 3); .......
No. 3.13. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (6; 1; -3); b (2; -4; 1); c (-1; –3; 4); d (15; 6; -17).