Option 7 DHS 2.1 Collection of DHS Ryabushko

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DHS - 2.1
No. 1.7. 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 = 3; β = 2; γ = -4; δ = -2; k = 2; ℓ = 5; φ = 4π / 3; λ = 1; μ = -3; ν = 0; τ = -1/2.
No. 2.7. 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 with respect to α :.
Given: A (1; 3; 2); B (–2; 4; –1); C (1; 3; –2); .......
No. 3.7. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (–3; 0; 1); b (2; 7; –3); c (–4; 3; 5); d (–16; 33; 13).

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