Option 30 DHS 2.1

DHS - 2.1
No. 1.30. 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 = -2; δ = 6; k = 4; ℓ = 7; φ = π / 3; λ = 2; μ = -1/2; ν = 3; τ = 2.
No. 2.30. 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 (4; 6; 7); B (2; –4; 1); C (- 3; –4; 2); .......
No. 3.30. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (–1; 4; 3); b (3; 2; –4); c (–2; –7; 1); d (6; 20; –3).