Option 17 IDS 2.1

IDZ – 2.1 No. 1.17. For given vectors $a = \alpha \cdot m + \beta \cdot n; b = \gamma \cdot m + \delta \cdot n; |m| = k; |n| = \ell; (m;n) = \varphi;$ must be found:

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

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

в) $\cos(a + \tau \cdot b).$

Дано: $\alpha = 5; \beta = -2; \gamma = 3; \delta = 4; k = 2; \ell = 5; \varphi = \pi/2; \lambda = 2; \mu = 3; \nu = 1; \tau = -2.$

No. 2.17. For vectors with coordinates of points $A, B$ and $C$, you need to find:

a) modulus of the 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 $\alpha.$

Hopefully: $A(4;5;3); B(-4;2;3); C(5;-6;-2).$

No. 3.17. It is necessary to prove that the vectors $a, b$ and $c$ form a basis, and find the coordinates of the vector $d$ in this basis.

Hopefully: $a(7;2;1); b(5;1;-2); c(-3;4;5); d(26;11;1).$

"Option 17 IDZ 2.1" is a digital product available for purchase in the digital goods store. This product contains solutions to problems from IPD 2.1 in linear algebra, including problems on calculating scalar products of vectors, projections of vectors and finding the coordinates of vectors in a given basis.

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IDZ 2.1 No. 1.17 is a task of finding some expressions using data from vectors and numerical coefficients. Vectors a and b are given as follows: a = 5m - 2n, b = 3m + 4n, |m| = 2, |n| = 5, (m;n) = π/2. The task consists of three points:

a) Find an expression for (λa + μb)·(νa + τb). The solution consists of substituting the given values, multiplying the vectors and adding the results.

b) Find the projection of the vector νa + τb onto the vector b. To solve this point, it is necessary to find the projection of the vector νa + τb onto the direction of the vector b, which is calculated as (νa + τb)·(b/|b|)·(b/|b|).

c) Find the value of cos(a + τb). To do this, you need to calculate the value of the scalar product of vectors a and b, as well as their length, and then apply the formula to find the cos angle between the vectors.

IDZ 2.1 No. 2.17 is a problem of calculating various characteristics of vectors given by the coordinates of points A, B and C. The given vectors are designated as a, b and c.

a) Find the modulus of vector a. This is calculated using the formula |a| = sqrt(a1^2 + a2^2 + a3^2), where a1, a2 and a3 are the coordinates of vector a.

b) Find the scalar product of vectors a and b. This is calculated by the formula a·b = a1b1 + a2b2 + a3b3, where a1, a2 and a3 are the coordinates of vector a, and b1, b2 and b3 are the coordinates of vector b.

c) Find the projection of vector c onto vector d. The projection of vector c onto vector d is calculated by the formula (c·d/|d|^2)·d, where c·d is the scalar product of vectors c and d, and |d|^2 is the square of the length of vector d.

d) Find the coordinates of the point M that divides the segment ℓ in relation to α. The coordinates of point M can be found using the formulas x = (1-α)A1 + αB1, y = (1-α)A2 + αB2, z = (1-α)A3 + αB3, where A1, A2, A3 are the coordinates of point A , B1, B2, B3 are the coordinates of point B, and x, y, z are the coordinates of point M.

IDZ 2.1 No. 3.17 is a task to find the coordinates of the vector d in the basis formed by the vectors a, b and c, and prove that these vectors form the basis.

a) Prove that the vectors a, b and c form a basis. To prove this, it is necessary to show that these vectors are linearly independent and that any vector can be expressed in terms of them by a linear combination.

b) Find the coordinates of vector d in the basis a, b and c. To do this, it is necessary to express vector d through a linear combination of vectors a, b and c, using a system of equations where the coefficients will be the desired coordinates. Then, by solving this system of equations, you can find the coordinates of the vector d in the basis of a, b and c.

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