IDZ Ryabushko 2.1 Variant 5

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 = 3; β = -2; γ = -4; δ = 5; k = 2; ℓ = 3; φ = π/3; λ = 2; μ = -3; ν = 4; 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( 2; 4; 5 ); В( 1; –2; 3 ); С(– 1; – 2; 4 ); ...

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; 1); b( –5; –3; 1 ); c( 2; –1; 0 ); d(– 15 ; –10 ; 5 ).