Version 2.1 Collection of 14 DHS DHS Ryabushko

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IDZ - 2.1
No. 1.14. 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 = -2; β = 3; γ = 5; δ = 1; k = 2; ℓ = 5; φ = 2π; λ = -3; μ = 4; ν = 2; τ = 3.
№ 2.14. 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:.
It is given: A (10; 6; 3); B (-2,3,5); C (3; -4; -6); .......
No. 3.14. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (4; 2; 3); b (-3; 1; -8); c (2; -4; 5); d (-12; 14; -31).

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