IDZ Ryabushko 2.1 Variant 2

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

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( 4 ; 3 ; – 2 ); В – 3 ; –1; 4 ); C( 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;4); b(–3;0;–2 ); c(4;5;–3 ); d(0;11;–14).