IDZ Ryabushko 2.1 Variant 15

No.1 Given a 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 + τb).
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 According to the coordinates of points A; B and C for the indicated vectors find: a) the module of the vector a;
b) the scalar product of the vectors a and b; c) the projection of the vector c onto the vector d; d) coordinates
points M; dividing the segment ℓ with respect to α :.
Given: А ( 3; 2; 4 ); В( – 2; 1; 3 ); С( 2; –2; –1); ...

No.3 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 ).