IDZ Ryabushko 2.1 Variant 6

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

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(– 1; –2; 4 ); В( –1; 3; 5 ); 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(3;1;2); b( –7; –2; –4 ); c(–4;0; 3); ( 16; 6; 15 ).