RAP IDZ 2.1 option 1

IHEID - 2.1
№ 1.1. The vectors are given by 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 + τ · b).
Given: α = -5; β = - 4; γ = 3; δ = 6; k = 3; ℓ = 5; φ = 5π / 3; λ = -2; μ = 1/3; ν = 1; τ = 2.
№ 2.1. From the coordinates of the points A; B and C for these vectors, find: a) the modulus of the vector a;
b) scalar product of vectors a and b; c) the projection of the vector c onto the vector d; d) coordinates
glasses M; dividing the segment ℓ with respect to α:.
Given: A (4; 6; 3); B (-5; 2; 6); C (4; -4; -3); .......
№ 3.1. 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).