IDZ Ryabushko 2.1 Variant 8

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 = 1; δ = -4; k = 3; ℓ = 2; φ = π; λ = 1; μ = - 2; ν = 3; 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( 2; –4; 3 ); В( –3; –2; 4 ); С( 0; 0; – 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( 5; 1; 2 ); b( –2; 1; –3 ); c( 4; –3; 5 ); d( 15; –15; 24 ).