Option 13 IDZ 2.2

No. 1.13. You are given the vectors a(-5;2;-2), b(7;0;-5) and c(2;3;-2). You need to do the following:

a) Calculate the mixed product of three vectors: (a x b) * c = ((20) - (3-5)) * -5 - ((-2*-5) - (2*-5)) * 3 + ((-5*-5) - (2*7)) * -2 = -35 - 30 + 9 = -56.

b) Find the modulus of the vector product: |a x b| = sqrt((2*-5 - 37)^2 + (-2-5 - (-5)(-5))^2 + (-52 - (-2)*(-5))^2) = sqrt(419).

c) Calculate the scalar product of two vectors: a * b = (-57) + (20) + (-2*-5) = -39.

d) Check whether two vectors are collinear or orthogonal: a and b are not collinear, since their mixed product is not equal to 0, and not orthogonal, since their scalar product is not equal to 0.

e) Check whether three vectors are coplanar: a, b and c are not coplanar, since their mixed product is not equal to 0.

No. 2.13. The tops of the pyramid are located at points A(-4;-7;-3), B(-4;-5;7), C(2;-3;3) and D(3;2;1).

No. 3.13. Given are three forces Q(9;-3;4), P(5;6;-2) and R(-4;-2;7) applied to point A(-5;4;-2). You need to do the following:

a) Calculate the work done by the resultant of these forces when the point of its application, moving rectilinearly, moves to point B(4;6;-5): for this you need to find the displacement vector from A to B, which is equal to B - A = (4 -(-5); 6-4; -5-(-2)) = (9;2;-3), and then calculate the scalar product of the resultant force and this displacement vector: W = Q * AB + P * AB + R * AB = (9*(-5+4) + (-3)(4-6) + 4(-2+5)) + (5*(-5+4) + 6*(4-6) + (-2)(-2+5)) + (-4(-5+4) + (-2)(4-6) + 7(-2+5)) = -51.

b) Calculate the magnitude of the resultant moment of these forces relative to point B: to do this, you need to find the vector from point B to point A, which is equal to A - B = (-5-4; 4-6; -2-(-5)) = ( -9;-2;3), and then calculate the vector product of the resultant force and this vector: M = AB x (Q+P+R) = (-9*(-10) - 2*(-9) + 3* (-3); (-3)(-10) - 4(-9) + 7*(-3); 4*(-9) - 5*(-2) - 9*(-4)) = (49;-11;19).

"Option 13 IDZ 2.2" is a digital product presented in a digital goods store with a beautiful html design. This product is intended for those who want to test their knowledge and skills in mathematics. The option includes tasks related to the calculation of mixed and vector products, scalar products and the determination of collinearity, orthogonality and coplanarity of vectors. The option also includes tasks related to calculating the work when moving the point of application of the resultant forces and the magnitude of the moment of the resultant forces relative to a given point.

This digital product is presented in a beautiful html design, which allows you to work with tasks conveniently and efficiently. Each task is designed as a separate block, which allows you to quickly switch between them and navigate the tasks. In addition, thanks to its convenient design, this digital product can be used both for independent work and for work in class or lectures.

If you want to test your knowledge and skills in mathematics or use this digital product for learning, “Option 13 IDZ 2.2” is an excellent choice for you.

"Option 13 IDZ 2.2" is a digital product presented in the digital goods store, which contains mathematics tasks related to the calculation of vector operations and working with forces.

In the first task, the vectors a(-5;2;-2), b(7;0;-5) and c(2;3;-2) are given. You need to calculate the mixed product of three vectors, which is -56, find the modulus of the cross product (sqrt(419)) and the dot product of the two vectors (–39), and also check whether the two vectors are collinear or orthogonal (a and b are not collinear and are not orthogonal) and check whether the three vectors are coplanar (a, b and c are not coplanar).

In the second task, the vertices of the pyramid in three-dimensional space are given: A(–4;–7;–3), B(–4; –5;7), C(2;–3;3) and D(3;2;1 ).

In the third task, three forces are given: Q(9;–3;4), P(5;6;–2) and R(–4;–2;7), applied to point A(–5;4;–2) . It is necessary to calculate the work produced by the resultant of these forces when the point of its application, moving rectilinearly, moves to point B(4;6;–5), which is equal to -51, as well as the magnitude of the moment of the resultant of these forces relative to point B, which is equal to the vector ( -9;-2;3), multiplied by the vector product AB and the sum of the vectors Q, P and R, that is (49;-11;19).

All tasks are designed in the form of separate blocks in a beautiful html design, which makes the use of this digital product convenient and effective both for independent work and for work in class or lectures.

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IDZ 2.2 is a task that includes several problems in linear algebra and mechanics.

In the first task, you need to perform several actions with the vectors a(-5;2;-2), b(7;0;-5) and c(2;3;-2):

• calculate the mixed product of three vectors;
• find the modulus of the vector product of a and b;
• calculate the scalar product of vectors a and b;
• check whether vectors a and b are collinear or orthogonal;
• check whether vectors a, b and c are coplanar.

In the second problem, you need to work with the vertices of the pyramid, given by the coordinates A(-4;-7;-3), B(-4;-5;7), C(2;-3;3) and D(3;2; 1).

In the third problem, you need to calculate the work done by the resultant of three forces Q(9;-3;4), P(5;6;-2) and R(-4;-2;7) applied to point A(-5; 4;-2), when the point of its application moves in a straight line to point B(4;6;-5). You also need to calculate the magnitude of the moment of the resultant of these forces relative to point B.

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