Option 25 DHS 2.1 Collection of DHS Ryabushko

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No. 1.25. Given the 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 + τ · b).
Given: α = 5; β = -8; γ = -2; δ = 3; k = 4; ℓ = 3; φ = 4π / 3; λ = 2; μ = -3; ν = 1; τ = 2.
No. 2.25. The coordinates of points A; B and C for the indicated vectors to find: a) the modulus of the vector a; b) the scalar product of vectors a and b; c) the projection of the vector c on the vector d; d) coordinates of the point M; dividing the segment ℓ in relation to α :.
Given: A (–5; 4; 3); B (4; 5; 2); C (2; 7; - 4); .......
No. 3.25. 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 (–4; 3; –1); c (2; 3; 4); d (14; 14; 20)

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