IDZ Ryabushko 2.1 Variant 1

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 = - 4; γ = 3; δ = 6; k is 3; ℓ = 5; φ = 5π / 3; λ = -2; μ = 1/3; 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: A (4; 6; 3); In (-5; 2; 6); C (4; –4; -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 (5; 4; 1); b (–3; 5; 2); c (2; –1; 3); d (7; 23; 4).