IDZ Ryabushko 2.1 Variant 9
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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 = -2; γ = 1; δ = 5; k = 3; ℓ = 6; φ = 4π/3; λ = -1; μ = 2; ν = 1; 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( 3; 4; –4);В( –2; 1; 2 ); С( 2; –3; 1); ...
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(0;2; –3); b( 4; – 3; –2 ); c( –5; –4; 0 ); d( –19; –5; –4 ).
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 = -2; γ = 1; δ = 5; k = 3; ℓ = 6; φ = 4π/3; λ = -1; μ = 2; ν = 1; 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( 3; 4; –4);В( –2; 1; 2 ); С( 2; –3; 1); ...
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(0;2; –3); b( 4; – 3; –2 ); c( –5; –4; 0 ); d( –19; –5; –4 ).
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