IDZ Ryabushko 2.1 Variant 18
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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 = 7; β = -3; γ = 2; δ = 6; k = 3; ℓ = 4; φ = 5π/3; λ = 3; μ = -1/2; ν = 2; 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( 2; 4; 6); В( –3; 5; 1 ); С( 4; –5; –4 ); ...
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(3;5;4); b(–2; 7;–5 );c(6;–2; 1); d( 6; –9; 22 ).
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 = 7; β = -3; γ = 2; δ = 6; k = 3; ℓ = 4; φ = 5π/3; λ = 3; μ = -1/2; ν = 2; 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( 2; 4; 6); В( –3; 5; 1 ); С( 4; –5; –4 ); ...
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(3;5;4); b(–2; 7;–5 );c(6;–2; 1); d( 6; –9; 22 ).
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