Option 8 DHS 2.1

Option 8 DHS 2.1

DHS - 2.1
№ 1.8. 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 = 5; β = 2; γ = 1; δ = -4; k = 3; ℓ = 2; φ = π; λ = 1; μ = - 2; ν = 3; τ = -4.
No. 2.8. 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 with respect to α :.
Given: A (2; –4; 3); B (–3; –2; 4); C (0; 0; -2); .......
No. 3.8. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (5; 1; 2); b (–2; 1; –3); c (4; –3; 5); d (15; –15; 24).