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No. 1 Given vectors a(4;2;-3), b(2;0;1) and c(-12;-6;9). It is necessary: a) calculate the mixed product of three vectors; b) find the modulus of the vector product; c) calculate the scalar product of two vectors; d) check whether two vectors are collinear or orthogonal; e) check whether the three vectors are coplanar.
No. 2 The tops of the pyramid are located at points A(7;5;8), B(–4;–5;3), C(2;–3;5) and D(5;1;–4).
No. 3 Force F(–9;5;7) is applied to point A(1;6;–3). It is necessary to calculate: a) the work of force in the case when the point of its application, moving rectilinearly, moves to point B(4;–3;5); b) modulus of the moment of force relative to point B.
"IDZ Ryabushko 2.2 Option 8" is a digital product intended for students studying mathematics in higher education institutions. This product contains several math activities that will help students improve their knowledge and skills in this area.
The design of the product is made in a beautiful HTML format, which makes it easier to perceive information and helps you quickly navigate tasks. In addition, this product is digital, which allows you to quickly and easily access materials without leaving your home.
"IDZ Ryabushko 2.2 Option 8" is an excellent choice for students who want to improve their knowledge in mathematics with the help of high-quality and convenient educational materials.
IDZ Ryabushko 2.2 Option 8 is a set of tasks in mathematics for students of higher educational institutions, presented in a convenient HTML format. Each task is accompanied by a detailed description and instructions for solving it.
No. 1 Given vectors a(4;2;-3), b(2;0;1) and c(-12;-6;9). Necessary: a) calculate the mixed product of three vectors; b) find the modulus of the vector product; c) calculate the scalar product of two vectors; d) check whether two vectors are collinear or orthogonal; e) check whether the three vectors are coplanar.
No. 2 The tops of the pyramid are located at points A(7;5;8), B(–4;–5;3), C(2;–3;5) and D(5;1;–4). Necessary:
No. 3 Force F(–9;5;7) is applied to point A(1;6;–3). You need to calculate: a) the work of force in the case when the point of its application, moving rectilinearly, moves to point B(4;–3;5); b) modulus of the moment of force relative to point B.
All tasks are classic examples from mathematics and will help improve students' knowledge and skills in this area. Thanks to the convenient format for perceiving information and the ability to access materials at any time, Ryabushko IDZ 2.2 Option 8 is an excellent choice for students who want to improve their knowledge in mathematics.
"IDZ Ryabushko 2.2 Option 8" is a digital product for students studying mathematics in higher education institutions. This product contains several math activities that will help students improve their knowledge and skills in this area.
In task No. 1, the vectors a(4;2;-3), b(2;0;1) and c(-12;-6;9) are given. It is necessary to complete the following tasks: a) calculate the mixed product of three vectors; b) find the modulus of the vector product; c) calculate the scalar product of two vectors; d) check whether two vectors are collinear or orthogonal; e) check whether the three vectors are coplanar.
In task No. 2, the vertices of the pyramid are given at points A(7;5;8), B(–4;–5;3), C(2;–3;5) and D(5;1;–4).
In task No. 3, force F(–9;5;7) is applied to point A(1;6;–3). It is necessary to calculate: a) the work of force in the case when the point of its application, moving rectilinearly, moves to point B(4;–3;5); b) modulus of the moment of force relative to point B.
The design of the product is made in a beautiful HTML format, which makes it easier to perceive information and helps you quickly navigate tasks. In addition, this product is digital, which allows you to quickly and easily access materials without leaving your home. If you are a student and want to improve your knowledge in mathematics with the help of high-quality and convenient educational materials, "IDZ Ryabushko 2.2 Option 8" is an excellent choice for you.
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IDZ Ryabushko 2.2 Option 8 is a linear algebra task that includes three tasks.
No. 1 Three vectors are given: a(4;2;-3) b(2;0;1) c(-12;-6;9)
a) It is necessary to calculate the mixed product of three vectors.
The mixed product of three vectors can be calculated using the formula: a (b x c) where x is the vector product symbol, · is the scalar product symbol.
Note that vector b and vector c are linearly dependent, since vector c = -3/2 * b. Therefore, the vector product b x c is equal to zero, and therefore the mixed product is equal to zero.
Answer: a · (b x c) = 0.
b) It is necessary to find the module of the vector product.
The modulus of the vector product of two vectors can be calculated using the formula: |b x c| = |b|*|c|*sin(angle between vectors).
Let's find the vector product of vectors b and c: b x c = (2;0;1) x (-12;-6;9) = (-6;-18;-12).
Let's find the module of the vector product: |b x c| = sqrt((-6)^2 + (-18)^2 + (-12)^2) = 6*sqrt(10).
Answer: |b x c| = 6*sqrt(10).
c) It is necessary to calculate the scalar product of two vectors.
The scalar product of two vectors can be calculated using the formula: a · b = |a|*|b|*cos(angle between vectors).
Let's find the scalar product of vectors a and b: a · b = (4;2;-3) · (2;0;1) = 8 - 3 = 5.
Answer: a · b = 5.
d) It is necessary to check whether two vectors are collinear or orthogonal.
Two vectors are collinear if they lie on the same line and have the same direction or opposite direction. Two vectors are orthogonal if their dot product is zero.
Let's calculate the scalar product of vectors a and b: a b = 5.
Since the scalar product is not zero, the vectors a and b are not orthogonal. To check for collinearity, you need to check whether these vectors are collinear to vector c.
Let's calculate the ratio of the coordinates of vectors a and c: 4/-12 = 2/-6 = -3/9.
The coordinate ratios of vectors a and c are not equal to each other, therefore, vectors a and c are not collinear.
Answer: Vectors a and b are neither orthogonal nor collinear.
d) It is necessary to check whether the three vectors are coplanar.
Three vectors are coplanar if they lie in the same plane. That is, if there is a vector that is orthogonal to each of them.
Let's calculate the vector product of vectors a and b: a x b = (4;2;-3) x (2;0;1) = (2;-10;-4).
Let's calculate the scalar product of vectors a x b and c: (a x b) · c = (2;-10;-4) · (-12;-6;9) = -24 + 60 - 36 = 0.
Since the scalar product is zero, the vector a x b is orthogonal to the vector c, which means all three vectors are coplanar.
Answer: Three vectors a, b and c are coplanar.
No. 2 The vertices of the pyramid are given: A(7;5;8), B(-4;-5;3), C(2,-3,5), D(5;1;-4).
You need to find the volume of the pyramid.
The volume of the pyramid can be calculated using the formula: V = 1/3 * S * h, where S is the area of the base of the pyramid, h is the height of the pyramid.
The area of the base can be calculated as the area of triangle ABC: S = 1/2 * |AB x AC|, where x is the vector product symbol, | | - vector module.
Calculate vectors AB and AC: AB = (-11;-10;-5), AC = (-5;-8;-3).
Compute the vector product AB x AC: AB x AC = (-10;20;-30).
Let's calculate the magnitude of the vector AB x AC: |AB x AC| = sqrt((-10)^2 + 20^2 + (-30)^2) = 10*sqrt(2)*sqrt(7).
So S = 1/2 * 10*sqrt(2)sqrt(7) = 5sqrt(14).
The height of the pyramid can be found as the distance from vertex D to the plane containing triangle ABC. To do this, find the equation of the plane passing through points A, B and C.
The plane normal vector can be found as the cross product of the vectors AB and AC: n = AB x AC = (-10;20;-30).
The equation of the plane is: -10x + 20y - 30z + D = 0, where D is an unknown constant that can be found by substituting the coordinates of point A: -107 + 205 - 30*8 + D = 0, D = 10.
Thus, the equation of the plane is: -10x + 20y - 30z + 10 = 0.
The height of the pyramid is equal to the distance from point D to the plane, which can be calculated using the formula: h = |(-105 + 201 - 30*(-4) + 10)| / sqrt((-10)^2 + 20^2 + (-30)^2).
Let's calculate the value: h = 9sqrt(14) / sqrt(1400) = 9sqrt(14) / 20.
So V = 1/3 * S * h = 1/3 * 5sqrt(14) * 9sqrt(14) / 20 = 27/4.
Answer: The volume of the pyramid is 27/4.
No. 3 Given are the force F(-9;5;7) applied to point A(1;6;-3) and point B(4;-3;5).
a) It is necessary to calculate the work of the force in the case when the point of its application, moving rectilinearly, moves to point B.
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Solution to problem 15.3.4 from the collection of Kepe O.E. - Рассмотрим задачу о бросании материальной точки массой m с поверхности Земли под углом ? = 60° к гор ... ...