IDZ Ryabushko 2.1 Variant 7

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 = -2; k = 2; ℓ = 5; φ = 4π/3; λ = 1; μ = -3; ν = 0; τ = -1/2.

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( 1; 3; 2); В(–2; 4; –1);С( 1; 3; –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; 0; 1); b( 2;7; –3); c(–4; 3; 5 ); d( –16; 33; 13 ).