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IDZ Ryabushko 2.1 Variant 27

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IDZ Ryabushko 2.1 Variant 27

No.1 Given a 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 + τb).
Given: α = -3; β = 4; γ = 5; δ = -6; k = 4; ℓ = 5; φ = π; λ = 2; μ = 3; ν = -3; τ = -1.

No.2 According to the coordinates of points A; B and C for the indicated vectors find: a) the module of the vector a;
b) the scalar product of the vectors a and b; c) the projection of the vector c onto the vector d; d) coordinates
points M; dividing the segment ℓ with respect to α :.
Given: А(6; 5; –4); В(–5;–2; 2);С( 3;3; 2); ...

No.3 Prove that the vectors a; b; c form a basis and find the coordinates of the vector d in this basis.
Given: a(4; 5; 1); b(1; 3; 1); c(–3; –6; 7); d(19; 33; 0).

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