IDZ Ryabushko 2.1 Variant 13

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 =4; β = 3; γ =-1; δ = 2; k = 4; ℓ = 5; φ = 3π/2; λ = 2; μ = - 3; ν = 1; τ = 2.
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(5;6;1); В( -5;2;6); С(3; –3 ;3 ); ...

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( 6;1;-3); b(2;-4;1); c(-1;–3;4); d(15;6;-17).