Solution IDZ Ryabushko 2.1 - Option 27

Section: Chapter 2. Vector algebra
Topic: Vectors and linear operations
Description: Linear operations on vectors, scalar product, projection of a vector, division of a segment in a given ratio, decomposition of a vector by basis.

Tasks:
1. Given vectors a=αm+βn and b=γm+δn, where |m|=k; |n|=l; (m,n)=φ. Find a) (λa+μb)(νa+τb); b) pr(νa+τb); c) cos(a,τb)
2. By coordinates of points A, B and C for the indicated vectors, find: a) the modulus of vector a; b) scalar product of vectors a and b; c) projection of vector c onto vector d; d) coordinates of point M dividing the segment l in the ratio α:β
3. Prove that vectors a,b,c form a basis and find the coordinates of vector d in this basis.

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