Version 2.1 Collection of 18 DHS DHS Ryabushko

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IDZ - 2.1
No. 1.18. 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 = 7; β = -3; γ = 2; δ = 6; k = 3; ℓ = 4; φ = 5π / 3; λ = 3; μ = -1/2; ν = 2; τ = 1.
№ 2.18. 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 (2; 4; 6); B (-3; 5; 1); C (4; -5; -4)
No. 3.18. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (3; 5; 4); b (-2; 7; -5); c (6; -2; 1); d (6; -9; 22)

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