Math Problem Statement

Выберите формулу для вычисления угла между двумя прямыми, заданными на плоскости общими уравнениями

Solution

The question asks you to select the formula for calculating the angle between two lines given in the plane by their general equations. In this context, the correct approach involves using the arctangent function with the appropriate combination of coefficients.

The general formula for the angle ϕ\phi between two lines given by the equations A1x+B1y+C1=0A_1x + B_1y + C_1 = 0 and A2x+B2y+C2=0A_2x + B_2y + C_2 = 0 is:

ϕ=arctan(A2B1A1B2A1A2+B1B2)\phi = \arctan \left( \frac{A_2B_1 - A_1B_2}{A_1A_2 + B_1B_2} \right)

Based on this formula, the correct answer is:

Option 2: ϕ=arctg(A2B1A1B2A1A2+B1B2)\phi = \arctg \left( \frac{A_2B_1 - A_1B_2}{A_1A_2 + B_1B_2} \right)

This option uses the arctangent (or "arctg" as written here) and matches the correct placement of the coefficients in the numerator and denominator.

Would you like further details or have any questions?

Here are some additional questions you may find useful to explore:

  1. How do you derive the angle formula for lines in the general form Ax+By+C=0Ax + By + C = 0?
  2. What is the meaning of each coefficient AA, BB, and CC in the line equation?
  3. How does the angle formula change for vertical or horizontal lines?
  4. Can this formula be applied to lines in three-dimensional space?
  5. What are some common applications of finding the angle between two lines?

Tip: When calculating angles between lines, always check if the lines are parallel (when the angle is 00^\circ or 180180^\circ) to simplify your solution.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Trigonometry

Formulas

Angle between two lines formula: \(\phi = \arctan \left( \frac{A_2B_1 - A_1B_2}{A_1A_2 + B_1B_2} \right)\)

Theorems

Formula for the angle between two lines

Suitable Grade Level

Grades 10-12