IDZ Ryabushko 2.1 Variant 3

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 = 5; β = -2; γ = -3; δ = -1; k = 4; ℓ = 5; φ = 4π/3; λ = 2; μ = 3; ν = -1; τ = 5.

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( –2; –2; 4 );В( 1; 3; –2 ); С( 1; 4; 2 ); ...

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(– 1; 1; 2); b( 2; –3; –5 ); c(–6; 3; –1 ); d( 28; –19; –7 ).