Option 18 DHS 2.1

DHS - 2.1
No. 1.18. Given the 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 + τ · b).
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. 2.18. The coordinates of points A; B and C for the indicated vectors to find: a) the modulus of the vector a; b) the scalar product of vectors a and b; c) the projection of the vector c on the vector d; d) coordinates of the point M; dividing the segment ℓ in relation to α :.
Given: A (2; 4; 6); B (–3; 5; 1); C (4; –5; –4)
No. 3.18. 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)