IDZ Ryabushko 2.1 Variant 11
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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 = -2; β =3; γ = 3; δ = -6; k =6;ℓ = 3; φ = 5π/3; λ = 3; μ = -1/3; ν = 1; 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(-2;-3;-4); В(2;-4;0); С(1;4;5); ...
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;3;1 ); b( –1;2;-3); c(3; –4;2); d(-9;34;-20).
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 = -2; β =3; γ = 3; δ = -6; k =6;ℓ = 3; φ = 5π/3; λ = 3; μ = -1/3; ν = 1; 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(-2;-3;-4); В(2;-4;0); С(1;4;5); ...
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;3;1 ); b( –1;2;-3); c(3; –4;2); d(-9;34;-20).
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