Option 20 IDZ 2.1

IDZ – 2.1 No. 1.20.

Given vectors a = α·m + β·n; b = γ m + δ n; |m| = k; |n| = ℓ; (m;n) = φ; Find:

а) ( λ·a + μ·b );( ν·a + τ·b );

b) projection ( ν·a + τ·b ) onto b;

в) cos( a + τ·b ).

Dano: α = 3; β = -5; γ =-2; d = 3; k = 1; ℓ = 6; φ = 3π/2; λ = 4; μ = 5; n = 1; τ = -2.

№ 2.20.

By coordinates of points A; B and C for the indicated vectors 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:

Hopefully: A(5;4;4 ); B(–5;2;3);C(4;2;– 5 ); …….

№ 3.20.

Prove that vectors a;b;c form a basis and find the coordinates of vector d in this basis.

Hopefully: a(11;1;2 ); b(–3;3;4); c(–4;–2; 7 ); d(–5; 11;–15 )

IDZ – 2.1 No. 1.20. For given vectors $a = \alpha\cdot m + \beta\cdot n$ and $b = \gamma\cdot m + \delta\cdot n$, where $|m| = k$, $|n| = \ell$, and $(m;n) = \varphi$, you need to find:

а) $(\lambda\cdot a + \mu\cdot b)\cdot(\nu\cdot a + \tau\cdot b)$;

b) projection of the vector $\nu\cdot a + \tau\cdot b$ onto the vector $b$;

в) $\cos(a + \tau\cdot b)$.

It is known that $\alpha = 3$, $\beta = -5$, $\gamma = -2$, $\delta = 3$, $k = 1$, $\ell = 6$, $\varphi = \frac{3\pi}{2}$, $\lambda = 4$, $\mu = 5$, $\nu = 1$, and $\tau = -2$.

No. 2.20. For vectors specified by the coordinates of points $A(5;4;4)$, $B(-5;2;3)$, and $C(4;2;-5)$, it is necessary 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 $\ell$ in relation $\alpha$.

No. 3.20. Prove that the vectors $a$, $b$, and $c$ form a basis and find the coordinates of the vector $d$ in this basis. It is known that $a(11;1;2)$, $b(-3;3;4)$, $c(-4;-2;7)$, and $d(-5;11;-15 ).

"Option 20 IDZ 2.1" is a digital product designed for students studying linear algebra. It contains a detailed description of the solution to three vector problems, as well as the given initial data needed to solve them. The product is designed in a beautiful html format, which ensures ease of reading and ease of perception of information. In addition, thanks to the digital format, the product can be purchased and downloaded at any convenient time and place, and also used on different devices without losing display quality. "Option 20 IDZ 2.1" is an excellent choice for students who want to deepen their knowledge in linear algebra and successfully complete assignments in this discipline.

Option 20 IDZ 2.1 is a digital product containing solutions to three problems in linear algebra.

In the first problem, vectors a and b are given, specified through their coefficients and basis vectors m and n, and the values ​​λ, μ, ν and τ are also given. It is necessary to find: a) the value of the scalar product of the vectors ( λ·a + μ·b ) and ( ν·a + τ·b ); b) projection of the vector ( ν·a + τ·b ) onto the vector b; c) the value of the cosine of the angle between vectors a and a + τ·b.

In the second problem, the coordinates of three points A, B and C are given, and you need to find: a) the magnitude of the vector given by the coordinates of points A and B; b) the scalar product of vectors specified by the coordinates of points A and B, and the coordinates of points A and C; c) projection of the vector specified by the coordinates of points C and D onto the vector specified by the coordinates of points A and B; d) coordinates of the point M dividing the segment AB in relation to α.

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

The product is designed in an easy-to-read HTML format and contains all the necessary initial data and a step-by-step description of how to solve problems. "Option 20 IDZ 2.1" is an excellent choice for students who want to deepen their knowledge in linear algebra and successfully complete assignments in this discipline.

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IDZ 2.1 No. 1.20 is a task that includes several points to be solved using these vectors and coefficients:

Given vectors a = α·m + β·n; b = γ m + δ n; |m| = k; |n| = ℓ; (m;n) = φ;

where α = 3; β = -5; γ =-2; d = 3; k = 1; ℓ = 6; φ = 3π/2;

Need to find:

а) ( λ·a + μ·b );( ν·a + τ·b );

b) projection ( ν·a + τ·b ) onto b;

в) cos( a + τ·b ).

To solve the problem you must complete the following steps:

1. Calculate vectors a and b using the given coefficients and vectors m and n.

2. Calculate λ·a + μ·b and ν·a + τ·b using these coefficients.

3. Find the scalar product ( λ·a + μ·b )·( ν·a + τ·b ).

4. Find the projection of the vector ν·a + τ·b onto the vector b.

5. Find cos( a + τ·b ) using the formula cos( a + τ·b ) = cos a · cos τ·b + sin a · sin τ·b.

The solution to the task might look like this:

1. a = 3·m - 5·n; b = -2·m + 3·n;

2. λ·a + μ·b = 4·a + 5·b = (12m - 20n) + (-10m + 15n) = 2m - 5n;

   ν·a + τ·b = 1·a - 2·b = (3m - 5n) - (-4m + 6n) = 7m - 11n;

3. ( λ·a + μ·b );( ν·a + τ·b ) = (2m - 5n)·(7m - 11n) = 14m^2 - 77mn + 55n^2;

4. The projection of the vector 7m - 11n onto the vector -2m + 3n is equal to ((7m - 11n)·(-2m + 3n))/(-2^2 + 3^2) = (-29m - 13n)/13;

5. cos( a + τ·b ) = cos a · cos τ·b + sin a · sin τ·b = ((3m - 5n)·(-2) + (2m - 3n)·1)/(√(9 +25·√(4+9)) = -11/13.

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