IDZ Ryabushko 2.2 Option 12

No. 1 Given vectors a(-4;3;-7), b(4;6;-2), c(6;9;-3). Necessary:

a) Calculate the mixed product of three vectors. To calculate the mixed product, it is necessary to calculate the determinant of a matrix composed of the coordinates of three vectors: a x b · c = | -4 3 -7 | | 4 6 -2 | | 6 9 -3 | = (-228; 6; 0)

b) Find the modulus of the vector product. To find the modulus of a vector product, it is necessary to calculate the length of the vector obtained as a result of the vector product of two vectors: |a x b| = |(-9; -38; 30)| = sqrt(9^2 + 38^2 + 30^2) ≈ 40.24

c) Calculate the scalar product of two vectors. To calculate the scalar product, it is necessary to multiply the corresponding coordinates of the vectors and add the resulting products: a · b = (-44) + (36) + (-7*-2) = 8 + 18 + 14 = 40

d) Check whether two vectors are collinear or orthogonal. Two nonzero vectors are collinear if one is a multiple of the other. Two nonzero vectors are orthogonal if their scalar product is zero. We calculate the scalar product of vectors a and b: a · b = 40 Since the scalar product is non-zero, vectors a and b are not orthogonal. We calculate the vector product of vectors a and b: a x b = (-9; -38; 30) If the vector product is zero, then the vectors are collinear. Since the cross product is nonzero, vectors a and b are not collinear.

e) Check whether the three vectors are coplanar. Three vectors are coplanar if they lie in the same plane. To check this condition, it is necessary to calculate the mixed product of three vectors. See answer to question a). If the mixed product is zero, then the vectors are coplanar. The mixed product of vectors a, b and c is not equal to zero, which means that these vectors do not lie in the same plane.

No. 2 The tops of the pyramid are located at points A(7;4;9), B(1;-2;-3), C(-5;-3;0), D(1;-3;4).

No. 3 Force F(2;2;9) is applied to point A(4;2;-3). Necessary:

a) Calculate the work done by force F in the case when the point of its application, moving rectilinearly, moves to point B(2;4;0). The work done by force F when moving the point of its application by vector d is equal to the scalar product of the force vector and the displacement vector: A = (4;2;-3), B = (2;4;0), F = (2;2;9) , d = B - A = (-2;2;3) W = F d = (2*-2) + (22) + (93) = 23

b) Calculate the modulus of the moment of force F relative to point B. The moment of force is equal to the product of the vector product of the radius vector and the force vector by the modulus of the radius vector: r = B - A = (-2;2;3), |r| = sqrt((-2)^2 + 2^2 + 3^2) = sqrt(17) M = |r x F| = |(2;-2;3) x (2;2;9)| = |(-24;-6;8)| = sqrt(24^2 + 6^2 + 8^2) ≈ 25.46

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IDZ Ryabushko 2.2 Option 12 is a task in linear algebra, which consists of three numbers.

No. 1. In this task you need to perform several operations with vectors. Given three vectors a(-4;3;-7), b(4;6;-2), c(6;9;-3). It is necessary to calculate the mixed product of three vectors, find the modulus of the cross product of vectors a and b, calculate the scalar product of vectors a and c, check whether vectors a and b are collinear or orthogonal, check whether three vectors a, b and c are coplanar.

No. 2. In this task you need to work with the vertices of the pyramid, defined by points A(7;4;9), B(1;-2;-3), C(-5;-3;0) and D(1;-3;4 ). It is necessary to find the volume of the pyramid defined by these points and the area of ​​its lateral surface.

No. 3. In this task, you need to calculate the work done by the force and the modulus of the moment relative to point B. Points A(4;2;-3) and B(2;4;0) are given, as well as the force F(2;2;9) applied to the point A. It is necessary to calculate the work done by force F if the point of its application moves from point A to point B, moving in a straight line, as well as the modulus of the moment of force F relative to point B.

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