Option 15 DHS 2.1

Option 15 DHS 2.1

DHS - 2.1
No. 1.15. 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 = 33; γ = 5; δ = 2; k = 4; ℓ = 7; φ = 4π / 3; λ = -3; μ = 2; ν = 2; τ = -1.
No. 2.15. 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 (3; 2; 4); B (- 2; 1; 3); C (2; –2; –1); .......
No. 3.15. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (- 2; 1; 3); b (3; –6; 2); c (-5; –3; –1); d (31; –6; 22)