IDZ Ryabushko 2.1 Option 12

No. 1. Need to find:

a) (λ·a + μ·b);(ν·a + τ·b);

b) projection (ν·a + τ·b) onto b;

в) cos(a + τ·b).

It is known that:

a = α·m + β·n;

b = γ·m + δ·n;

|m| = k;

|n| = ℓ;

(m;n) = φ;

α = -2;

β = -4;

γ = 3;

d = 6;

k = 3;

ℓ = 2;

φ = 7π/3;

λ = -1/2;

m = 3;

n = 1;

τ = 2.

No. 2. For vectors a, b, c you need to find:

a) modulus of vector a;

b) scalar product of vectors a and b;

c) projection of vector c onto vector d;

d) coordinates of the point M dividing the segment ℓ in relation to α.

The coordinates of the points A(-2;-3;-2), B(1;4;2), C(1;-3;3) and the vectors a, b, c are known.

No. 3. It is necessary to prove that vectors a, b, c form a basis, and find the coordinates of vector d in this basis.

The vectors a(3;1;-3), b(-2;4;1), c(1;-2;5) and the vector d(1;12;-20) are known.

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IDZ Ryabushko 2.1 Option 12 is a task in linear algebra, which consists of three numbers.

In the first issue, vectors a and b are given, as well as numerical values, and you need to find:

a) (λ·a + μ·b);(ν·a + τ·b);

b) projection (ν·a + τ·b) onto b;

в) cos(a + τ·b).

The second number consists of finding some characteristics of the vectors a, b, c and d from their coordinates. In particular, it is required:

a) find the modulus of vector a;

b) find the scalar product of vectors a and b;

c) find the projection of vector c onto vector d;

d) find the coordinates of the point M dividing the segment ℓ in the given relation α.

In the third issue, you need to prove that vectors a, b and c form a basis, and find the coordinates of vector d in this basis.

To do this, you need to solve a system of linear equations, where the coefficients of vectors a, b and c are unknown, equating the linear combination of these vectors to vector d.

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