Angle of Intersections of Curves

Determining the Angle of Intersection of Curves Using Vectors

Overview

  • We'll explore how to find the angle of intersection of two curves using vectors.

  • The curves we are examining are:

    • f(x) = x² - 4x + 4

    • g(x) = 3x + 4

  • The curves intersect at the point (0, 4).

  • Our objective is to calculate the angle of intersection in radians (from 0 to π) and round the answer to two decimal places.

Step 1: Finding the Derivatives

  • Find the derivatives of both functions:

    • f'(x) = d/dx (x² - 4x + 4) = 2x - 4

    • g'(x) = d/dx (3x + 4) = 3

  • Evaluate the derivatives at the intersection point (x = 0):

    • f'(0) = 2(0) - 4 = -4

      • Slope of tangent line to f(x) at intersection = -4

    • g'(0) = 3

      • Slope of tangent line to g(x) at intersection = 3

Step 2: Sketching the Tangent Lines

  • Tangent Line to f(x) at (0, 4):

    • Slope = -4 (down 4 units, right 1 unit).

    • Sketch line: It extends downwards due to the negative slope.

  • Tangent Line to g(x):

    • This is the same line as g(x) since it’s linear with a constant slope of 3.

  • Angles of Intersection:

    • An obtuse angle and an acute angle present at the intersection.

Step 3: Forming Vectors from Slopes

  • Define Vector V (for f'(0)):

    • Slope = -4 (slope can be written as -4/1).

    • Start at (0, 4): move down 4 units & right 1 unit → (1, 0).

    • Vector V = (1, -4).

  • Define Vector W (for g'(0)):

    • Slope = 3 (slope can be written as 3/1).

    • Start at (0, 4): move up 3 units & right 1 unit → (1, 7).

    • Vector W = (1, 3).

Step 4: Writing Vectors in Component Form

  • Vector V:

    • Components: (1, -4)

    • Slope check: -4 / 1 = -4

  • Vector W:

    • Components: (1, 3)

    • Slope check: 3 / 1 = 3

Step 5: Calculating the Angle of Intersection

  • Angle Calculations:

    • Use the formula:[ cos(\theta) = \frac{\text{Dot Product of } V and W}{\text{Magnitude of V} \times \text{Magnitude of W}} ]

    • Dot Product:[ V \cdot W = (1)(1) + (-4)(3) = 1 - 12 = -11 ]

    • Magnitudes:

      • Magnitude of V: [ ||V|| = \sqrt{(1)² + (-4)²} = \sqrt{1 + 16} = \sqrt{17} ]

      • Magnitude of W: [ ||W|| = \sqrt{(1)² + (3)²} = \sqrt{1 + 9} = \sqrt{10} ]

  • Final Calculation for Cosine of Angle:[ cos(\theta) = \frac{-11}{\sqrt{17} \times \sqrt{10}} ]

Conclusion

  • This allows us to calculate the angle in radians between the two curves at the intersection point.

  • Remember that one angle will be acute and the other obtuse.

  • Continue to evaluate and find both angles by further solving for ( \theta ).