Math Problem Statement

Given vectors a = (5, -1, 2) and b = -i + j, find the scalar product 2a · (a - 2b).

Solution

The image contains mathematical problems in Russian. Let me transcribe and analyze the visible part of the text to provide solutions.

Transcription:

The visible part includes the following:

  1. Problem 2: "Найти модуль вектора" (Find the magnitude of the vector).
    This problem involves finding the magnitude of a given vector (not completely visible).

  2. Problem 3:
    Given: a=(5,−1,2)\mathbf{a} = (5, -1, 2), b=−i+j\mathbf{b} = -\mathbf{i} + \mathbf{j}.
    Task: Find the scalar product:
    2a⋅(a−2b).2\mathbf{a} \cdot (\mathbf{a} - 2\mathbf{b}).

Solution for Problem 3:

To solve 2a⋅(a−2b)2\mathbf{a} \cdot (\mathbf{a} - 2\mathbf{b}), we break it into steps:

Step 1: Represent vectors.

  • a=(5,−1,2)\mathbf{a} = (5, -1, 2).
  • b=−i+j=(−1,1,0)\mathbf{b} = -\mathbf{i} + \mathbf{j} = (-1, 1, 0).

Step 2: Compute a−2b\mathbf{a} - 2\mathbf{b}.

= (5 + 2, -1 - 2, 2 - 0) = (7, -3, 2).$$ #### Step 3: Compute $$2\mathbf{a}$$. $$2\mathbf{a} = 2(5, -1, 2) = (10, -2, 4).$$ #### Step 4: Find $$2\mathbf{a} \cdot (\mathbf{a} - 2\mathbf{b})$$ (scalar product). The scalar product is: $$2\mathbf{a} \cdot (\mathbf{a} - 2\mathbf{b}) = (10, -2, 4) \cdot (7, -3, 2).$$ $$= (10 \cdot 7) + (-2 \cdot -3) + (4 \cdot 2) = 70 + 6 + 8 = 84.$$ --- ### Final Answer for Problem 3: $$\boxed{84}$$ Would you like a detailed explanation of Problem 2 or further clarifications? Let me know! --- ### Related Questions: 1. What are the properties of scalar and vector products? 2. How do you compute the magnitude of a vector in 3D space? 3. Can this approach be extended to higher dimensions (4D, 5D, etc.)? 4. What are the geometric interpretations of scalar and vector products? 5. How does changing the scalar coefficient (e.g., $$2\mathbf{a}$$) affect the scalar product? ### Tip: Always double-check calculations in vector problems, as small sign errors can lead to incorrect results!

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Math Problem Analysis

Mathematical Concepts

Vector Operations
Scalar Product
Linear Algebra

Formulas

Magnitude of a vector: ||v|| = √(x^2 + y^2 + z^2)
Scalar product: u · v = u1v1 + u2v2 + u3v3
Distributive property of scalar products

Theorems

Properties of Scalar Products

Suitable Grade Level

Grades 10-12

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