Option 8 2.1 Collection DHS DHS Ryabushko

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IDZ - 2.1
No. 1.8. 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 = 2; γ = 1; δ = -4; k = 3; ℓ = 2; φ = π; λ = 1; μ = - 2; ν = 3; τ = -4.
No. 2.8. 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 of the dividing segment ℓ with respect to α:.
Given: A (2; -4; 3); B (-3; -2; 4); C (0; 0; -2); .......
No. 3.8. Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a (5; 1; 2); b (-2; 1; -3); c (4; -3; 5); d (15; -15; 24).

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