IDZ Ryabushko 2.1 Variant 29
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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 = 5; β=3; γ = – 4; δ = –2; k = 6; ℓ = 3; φ = 5π/3; λ= –2; μ= –1/2; ν = 3; τ= 2.
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; 5; 4 );В( 4; 2; –3 ); С(–2; 4; 7 ); ...
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( 5;7; –2 ); b(–3;1;3); c(1; –4; 6); d( 14; 9; – 1).
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 = 5; β=3; γ = – 4; δ = –2; k = 6; ℓ = 3; φ = 5π/3; λ= –2; μ= –1/2; ν = 3; τ= 2.
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; 5; 4 );В( 4; 2; –3 ); С(–2; 4; 7 ); ...
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( 5;7; –2 ); b(–3;1;3); c(1; –4; 6); d( 14; 9; – 1).
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