Version 2.1 Collection of 10 DHS DHS Ryabushko

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IDZ - 2.1
No. 1.10. 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 = -3; γ = 4; δ = 2; k = 4; ℓ = 1; φ = 2π / 3; λ = 2; μ = -1 / 2; ν = 3; τ = 0.
№ 2.10. 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) the coordinates of the point M; dividing the segment ℓ with respect to α:.
Given: A (0; 2; 5); In (2; -3; 4); C (3; 2; -5); .......
№ 3.10. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (3; -1; 2); b (-2,3; 1); c (4; -5; -3); d (-3; 2; -3).

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